University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


A  SCHOOL  ALGEBRA 
COMPLETE 


FLETCHER  DUEELL,  Ph.D. 

Mathematical  Master  in  the  Lawbencevillb  School 

AND 

EDWARD    R.    ROBBINS,   A.B. 

Mathematical  Masteb  in  the  William  Penn  Chaeteb  School 


NEW  YORK 

CHARLES    E.    MERRILL  CO. 


Durell  &  Robbins' 
Mathematical  Series 


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a 


Copyright  1897,   by  Charles  E.  Merrill  Co. 


PREFACE. 


The  principal  object  in  writing  this  School  Algebra  has 
been  to  simplify  principles  and  make  them  attractive,  by 
showing  more  plainly,  if  possible,  than  has  been  done  here- 
tofore, the  practical  or  common-sense  reason  for  each  step 
or  process.  Thus,  at  the  outset  it  is  shown  that  new 
symbols  are  introduced  into  algebra  not  arbitrarily,  but  for 
the  sake  of  definite  advantages  in  representing  numbers. 
The  fundamental  laws  of  algebra  governing  the  use  of  sym- 
bols derive  their  importance  in  like  manner  from  the  econ- 
omies which  they  make  possible  in  dealing  with  the  symbols 
for  numbers.  Each  successive  process  is  taken  up  for  the 
sake  of  the  economy  or  new  power  which  it  gives  as  com- 
pared with  previous  processes. 

It  is  hoped  that  this  treatment  not  only  makes  each  prin- 
ciple clearer  to  the  pupil,  but  also  gives  increased  unity  to 
the  subject  as  a  whole.  It  is  also  believed  that  this  treatment 
of  algebra  is  better  adapted  to  the  practical  American  spirit, 
and  gives  the  study  of  the  subject  a  larger  educational  value. 

While  seeking  to  develop  the  theory  of  the  subject  in  this 
manner,  it  has  been  deemed  best  to  keep  in  close  touch  with 
the  best  current  practice  of  teachers  in  other  respects.  For 
instance,  the  order  of  topics  in  text-books  most  used  at  pres- 
ent has  been  followed. 

8 


4  PREFACE. 

Great  care  has  been  taken  in  the  selection  and  gradation 
of  a  large  number  of  examples.  It  is  hoped  that  they  have 
been  so  graded  that  any  example  may  be  considered  the  last 
of  a  series  of  progressive  steps,  provided  the  teacher  wishes 
to  limit  the  work  at  any  particular  point.  Frequent  reviews 
have  been  provided  for,  especially  in  the  all-important  sub- 
jects of  Factoring,  Fractions,  Exponents,  and  Radicals. 

This  volume  contains,  besides  the  specified  requirements  in 
algebra  for  admission  to  the  classical  course  of  colleges,  the 
more  advanced  subjects  required  by  universities  and  scien- 
tific schools — to  wit,  Permutations  and  Combinations,  Unde- 
termined Coefficients,  The  Binomial  Theorem,  Continued 
Fractions,  and  Logarithms. 

The  authors  will  sincerely  appreciate  the  courtesy,  if  their 
friends  and  fellow-teachers  will  kindly  advise  them  of  any 
discovered  errors. 


FLETCHER  DURELL, 
EDWARD  R.  ROBBINS. 


Lawrencevill,e,  N.  J.,  1 
December  23,  1897.      / 


CONTENTS. 


CHAPTER  I. 

PAGE 

ALGEBRAIC  SYMBOLS 9 

I.  Symbols  of  Quantity 10 

II.  Symbols  of  Operation 12 

III.  Symbols  of  Relation 14 

Algebraic  Expressions 15 

CHAPTER  11. 

METHODS  OF  USING  ALGEBRAIC  SYMBOLS .20 

I.  Laws  for  -f-  and  —  Signs 20 

II.  Laws  of  Arrangement  and  Grouping 22 

CHAPTER  HI. 

ADDITION  AND  SUBTRACTION ! 26 

Addition 26 

Subtraction 29 

Use  of  Parenthesis 32 

CHAPTER  IV. 

MULTIPLICATION 38 

Multiplication  of  Monomials 38 

Polynomial  by  a  Monomial 40 

Polynomial  by  a  Polynomial 41 

CHAPTER  V. 

DIVISION 48 

Division  of  Monomials 48 

Polynomial  by  a  Monomial 50 

Polynomial  by  a  Polynomial 51 

5 


6  CONTENTS.  * 

CHAPTER  VI. 

PAGE 

SIMPLE  EQUATIONS 57 

Solution  of  Problems 63 

CHAPTER  VII. 

ABBREVIATED  MULTIPLICATION  AND  DIVISION  ....  73 

Abbreviated  Multiplication 73 

Abbreviated  Division 81 

CHAPTER  VIII. 
FACTORING 86 

CHAPTER  IX. 

HIGHEST    COMMON    FACTOR    AND    LOWEST    COMMON 
MULTIPLE ]04 

CHAPTER  X. 

FRACTIONS 117 

General  Principles 118 

Transformations  of  Fractions 119 

Processes  with  Fractions  .   .   .....   . 126 

CHAPTER  XL 

FRACTIONAL  AND  LITERAL  EQUATIONS 143 

Problems 153 

CHAPTER  XII. 
SIMULTANEOUS  EQUATIONS .163 

CHAPTER  XIII. 

PROBLEMS    INVOLVING    TWO     OR     MORE    UNKNOWN 
QUANTITIES 178 

CHAPTER  XIV. 
INEQUALITIES ....!....   188 


CONTENTS.  7 

CHAPTER  XV. 

PAGX 

INVOLUTION  AND  EVOLUTION 192 

Powers  of  Monomials 192 

Powers  of  Binomials 194 

Evolution  .... 197 

Square  Root 199 

Cube  Root  ....... 206 

CHAPTER  XVI. 

EXPONENTS 213 

Fractional  Exponents 214 

Negative  Exponents 216 

Polynomials,  etc.       223 

CHAPTER  XVIL 

RADICALS 226 

Transformations  of  Radicals 227 

Operations  with  Radicals     .   . ?  r  •  •232 

CHAPTER  XVIII. 
IMAGINARIES 261 

CHAPTER  XIX. 

QUADRATIC  EQUATIONS  OF  ONE  UNKNOWN  QUANTITY  259 

Pure  Quadratic  Equations 259 

Affected  Quadratic  Equations 261 

Factorial  Method  of  Solving  Equations  ........   265 

Equations  in  the  Quadratic  Form 266 

Radical  Equations  . .   .   ;   -  .   •  •   •  269 

Other  Methods  of  Solving  Quadratic  Equations  ....  270 

CHAPTER  XX. 

SIMULTANEOUS  QUADRATIC  EQUATIONS    ........   278 

Special   Methods   of   Solving   Simultaneous  Quadratic 
Equations , 28X 


8  CONTENTS. 

CHAPTER  XXI. 
GENERAL  PROPERTIES  OF  QUADRATIC  EQUATIONS.   .   289 
Properties  of  ax^  +  bx  +  c  =  0 292 

CHAPTER  XXII. 

RATIO  AND  PROPORTION 295 

Ratio    . 295 

Proportion 296 

CHAPTER  XXIII. 

INDETERMINATE  EQUATIONS.    VARIATION 304 

Variation 307 

Kinds  of  Elementary  Variations 308 

CHAPTER  XXIV. 
ARITHMETICAL  PROGRESSION 314 

CHAPTER  XXV. 
GEOMETRICAL  AND  HARMONICAL  PROGRESSIONS  ...   323 
Harmonical  Progression •   •   • 333 

CHAPTER  XXVI. 
PERMUTATIONS  AND  COMBINATIONS 336 

CHAPTER  XXVII. 

UNDETERMINED  COEFFICIENTS 344 

I.  Expansion  of  a  Fraction  into  a  Series 347 

II.  Expansion  of  a  Radical  into  a  Series 349 

III.  Partial  Fractions 350 

IV.  Reversion  of  Series 355 

CHAPTER  XXVIII. 
THE  BINOMIAL  THEOREM 358 

CHAPTER  XXIX. 
CONTINUED  FRACTIONS 367 

CHAPTER  XXX. 
LOGARITHMS 376 

CHAPTER   XXXI. 
HISTORY  OF  ELEMENTARY  ALGEBRA 393 

CHAPTER  XXXII. 
APPENDIX 403 


SCHOOL  ALGEBRA  COMPLETE. 


CHAPTER    I. 
ALGEBRAIC  SYMBOLS. 

1.  First  Source  of  New  Power  in  Algebra.  Let  the  fol- 
lowing problem  be  proposed  for  solution : 

James,  John,  and  William  together  have  120  marbles.  John 
has  three  times  as  many  marbles  as  James,  and  William  has 
twice  as  man}^  as  John.     How  many  marbles  has  each  boy  ? 

The  solution  of  the  problem  is  facilitated  by  the  use  of 
some  symbol,  as  x,  for  one  of  the  unknown  numbers.     Thus, 

Let  X  =  number  of  marbles  which  James  has, 

thenSx  =       "  "  "      John  has, 

6x  =       "  "  "      William  has. 

Hence,  x  +  'Sx  +  Qx=       "  "  "      all   have   to- 

gether. 
But  120=       "  "  "      all   have   to- 

gether. 
Hence,  x  +  Sx  +  6x  =  120 
that  is,  lOc  =  120 

x  —  12,  number  of  marbles  which  James  has, 
3a:  =  36,       "  "  "      John      " 

6a;  =  72,       "  "  "      William" 

This  solution  of  the  problem  illustrates  the  first  new  prin- 
ciple in  algebra — viz.  that  a  more  extended  use  of  symbols 
than  is  practised  in  arithmetic  gives  increased  ease  and  power 
in  the  investigation  of  properties  of  number.     In  the  above 


10  ALGEBRA. 

example,  a  symbol  being  used  for  one  unknown  number,  the 
other  unknown  numbers  may  be  expressed  in  terms  of  this 
symbol ;  then  the  relation  of  all  the  unknown  numbers  to  the 
known  number  is  expressed  in  a  form  which  is  readily  reduced 
to  another  form  so  simple  that  from  it  the  value  of  the  first 
unknown,  and  afterward  the  values  of  the  other  unknown 
numbers  are  at  once  perceived. 

2.  Definition  of  Algebra.  Hence,  Algebra,  in  its  first  con- 
ception, is  that  branch  of  mathematics  which  treats  of  the 
properties  of  number  (or  quantity  expressed  by  number)  by 
the  extended  use  of  symbols. 

Algebra  may  be  briefly  described  as  generalized  arithmetic. 

3.  Three  Classes  of  Symbols.  Three  principal  kinds  of 
symbols  are  used  in  algebra: 

I.  Symbols  of  Quantity. 
II.  Symbols  of  Operation. 
III.  Symbols  of  Relation. 

I.  Symbols  op  Quantity. 

4.  Symbols  for  Known  Quantities.  Known  quantities  ar« 
represented  in  arithmetic  by  figures ;  as,  2,  3,  27,  etc.  They 
are  represented  in  the  same  way  in  algebra,  but  also  in  an- 
other more  general  way — viz.  by  the  first  letters  of  the  alpha- 
bet ;  as,  a,  6,  c,  etc. 

The  advantage  in  the  use  of  letters  to  represent  known 
numbers  lies  in  the  fact  that  a  letter  may  stand  for  any 
known  number,  and  a  result  be  obtained  by  ihe  use  of 
letters  which  is  true  for  all  numbers. 

5.  Symbols  for  Unknown  Quantities.  Unknown  quan^ 
titles  in  algebra  are  usually  denoted  by  the  hist  letters  of  tlib 
alphabet ;  as,  x,  y,  z,  w,  v,  etc. 

The  advantage  in  the  use  of  a  distinct  symbol  for  an  un- 
known quantity  is  stated  in  Art.  1. 


ALGEBRAIC  SYMBOLS.  IV 

6.  Symbols  for  Groups  of  Similar  Quantities.  Groups 
of  similar  quantities  are  usually  represented  by  groups  of 
similar  symbols;  as, 

(1)  By  the  same  letter  with  different  accents;  for  example, 
a',  a",  a!'\  etc.,  read  "  a  prime,"  "  a  second,"  "  a  third,"  etc. 

(2)  By  the  same  letter  with  different  subscript  figures ;  as, 
«!,  «2,  CL-ii  etc.,  read  "  a  sub-one,"  "  a  sub-two,"  etc. 

The  advantages  of  these  ways  of  representing  groups  of 
similar  quantities  are  obvious. 

7.  Sign  of  Continuation.  After  a  group  of  similar  quan- 
tities a  series  of  dots  is  often  written  to  indicate  that  the 
group  is  continued  indefinitely. 

Ex.  tti,  a2,  as, 

This  series  of  dots  is  called  the  sign  of  continuation,  and 
reads  "  and  so  on." 

8.  Positive  and  Negative  Quantity.  Negative  quantity 
is  quantity  exactly  opposite  in  quality  or  condition  to  quan- 
tity taken  as  positive. 

If  distance  east  of  a  certain  point  is  taken  as  positive,  dis- 
tance west  of  that  point  is  called  negative. 

If  north  latitude  is  positive,  south  latitude  is  negative. 

If  temperature  above  zero  is  taken  as  positive,  temperature 
below  zero  is  negative. 

If  in  business  matters  a  man's  assets  are  his  positive  pos- 
sessions, his  debts  are  negative  quantity. 

Positive  and  negative  quantity  are  distinguished  by  the 
signs  +  and  —  placed  before  them.  Thus,  $50  assets  are 
denoted  by  +$50,  and  $30  debts  by  -$30.  We  de- 
note 12°  above  zero  by  + 12°,  and  10°  below  zero  by 
- 10°. 

The  use  of  the  symbols  +  and  -  for  this  purpose,  as 
well  as  to  indicate  the  operations  of  addition  and  subtrac- 
tion (see  Arts.  9  and  10),  will  be  justified  later  on  (see  Arts. 
31,32). 


12  ALGEBRA. 

n.  Symbols  of  Operation. 

9.  The  Sign  of  Addition  is  -f-,  and  is  called  "plus." 
Placed  between  two  quantities,  it  indicates  that  the  quan- 
tity after  the  plus  sign  is  to  be  added  to  the  quantity 
before  it. 

Thus,  a  +  &  is  read  "a  plus  6,"  and  indicates  that  the  quan- 
tit}^  b  is  to  be  added  to  the  quantity  a. 

10.  The  Sign  of  Subtraction  is  — ,  and  is  called  "  minus." 
Placed  between  two  quantities,  it  indicates  that  the  second 
quantity  is  to  be  subtracted  from  the  first. 

Thus,  a  —  6  is  read  "  a  minus  6,"  and  means  that  b  is  to  be 
subtracted  from  a. 

11.  The  Sign  of  Multiplication  is  X,  and  reads  "times" 
or  "  multiplied  by,"  or  simply  "  into."  Placed  between  two 
quantities,  it  indicates  that  the  one  is  to  be  multiplied  by  the 
other. 

Thus,  aXb  reads  "  a  multiplied  by  6,"  and  means  that  a 
and  b  are  to  be  multiplied  together. 

The  multiplication  of  literal  quantities  (and  sometimes  of 
arithmetical  numbers)  may  also  be  indicated  more  simply  by 
a  dot  placed  between  the  quantities.  The  multiplication  of 
literal  quantities  is  indicated  most  simply  of  all  by  the  omis- 
sion of  any  symbol  between  the  quantities. 

Thus,  instead  of  a  X  6,  we  may  WTite  a-b,  or  ab. 

12.  Factors.  The  factors  of  a  number  are  the  numbers 
which  multiplied  together  produce  the  given  number. 

For  example,  the  factors  of  14  are  7  and  2 ;  the  factors  of 
abc  are  a,  b,  and  c. 

13.  Coefficients.  In  case  a  numerical  factor  occurs  in  a 
product,  it  is  written  first,  and  is  called  a  coefficient.     Hence, 

A  Coefficient  is  a  number  prefixed  to  a  given  quantity  to 
show  how  many  times  the  given  quantity  is  taken. 


ALGEBRAIC  SYMBOLS.  13 

For  example,  in  bxy^  5  is  called  the  coefficient.  When  the 
coefficient  is  1,  the  1  is  not  written,  but  is  understood. 

Thus,  xy  means  \xy. 

When  a  number  of  factors  are  multiplied  together,  any  one 
of  them  or  the  product  of  any  number  of  them  may  be  re- 
garded as  the  coefficient. 

Thus,  in  bahx^  bah  is  sometimes  regarded  as  the  coeffi- 
cient. 

14.  Powers  and  Exponents.  A  Power  is  the  product  of 
a  number  of  equal  factors. 

The  expression  for  a  power  is  abbreviated  by  the  use  of  the 
exponent. 

An  Exponent  is  a  small  figure  or  letter  written  above  and 
to  the  right  of  a  quantity  to  indicate  how  many  times  the 
quantity  is  taken  as  a  factor. 

Thus,  for  xxxx,  or  four  x's  multiplied  together,  we  write  a;*, 
the  exponent  in  this  ca-se  being  4. 

An  exponent  is  thus  in  effect  a  symbol  of  operation. 

When  the  exponent  is  unity  it  is  omitted.  Thus,  x  is  used 
instead  of  ic\  and  means  x  to  the  first  power. 

15.  The  Sign  of  Division  is  ^,  and  reads  "  divided  by." 
Placed  between  two  quantities,  it  indicates  that  the  quantity 
to  the  left  of  the  sign  is  to  be  divided  by  the  quantity  to  the 
right  of  it. 

Thus,  a  H-  6  means  that  a  is  to  be  divided  by  h. 
Division  may  also  be  indicated  by  placing  the  quantity  to 
be  divided  above  a  horizontal  line  and  the  divisor  below  the 

line.     Thus,  for  an- 6  we  may  write  -• 

The  expression  -  is  often  read  "  a  over  6." 
h 

16.  The  Radical  Sign  is  ;/,  and  means  that  the  root  of 
the  quantity  following  it  is  to  be  extracted.  The  degree  of 
the  root  is  indicated  by  a  small  figure  placed  above  the  rad- 


t4  ALGEBRA. 

ical  sign.     For  the  square  root  the  figure  or  index  of  the  root 
is  omitted. 

Thus,  Vd  means  "square  root  of  9." 
Va  means  "  cube  root  of  a." 

m.  Symbols  of  Relation. 

17.  The  Sign  of  Equality  is  =,  and  reads  "  equals  "  or  "  is 
equal  to." 

When  placed  between  two  quantities  it  indicates  that  they 
are  equal  to  each  other. 

Thus,  a  =  b  means  that  a  and  b  are  equal  quantities. 

18.  The  Signs  of  Inequality  are  >,  which  reads  "is  greater 
than,"  and  <,  which  reads  "  is  less  than." 

Thus,  a  >  6  means  that  a  is  greater  than  b. 

c  <ib  means  that  c  is  less  than  b. 
It  is  to  be  observed  that  in  both  the  signs  of  inequality  tha 
opening  is  toward  the  greater  quantity. 

19.  The  Signs  of  Aggregation  are  the  parenthesis  (  ),  th& 
brackets  [  ],  the  braces  \  j,  and  the  vinculum  . .  Any 
one  of  these  indicates  that  all  the  quantities  inclosed  b}^  it 
are  to  be  treated  as  a  single  quantity ;  that  is,  subjected  to 
the  same  operation. 

Thus,  5(2a  —  b  +  c)  means  that  all  the  quantities  inside  the 
parenthesis — viz.  2a,  —  6,  +  c — are  each  to  be  multiplied  by  5. 

Again,  (a  +  2b)  (a  +  26  +  c)  means  that  the  sum  of  thb 
quantities  in  the  first  parenthesis  is  to  be  multiplied  by  the 
sum  of  those  in  the  second  parenthesis. 

The  sign  of  aggregation  ordinarily  used  is  the  parenthesis ; 
the  other  symbols  of  aggregation  are  used  in  cases  where  con- 
fusion might  result  if  several  parentheses  were  used  together. 

20.  The  Sign  of  Deduction  is  . " . ,  and  is  read  "  therefore  " 
or  "  hence." 

It  is  used  to  show  the  relation  between  succeeding  proposi- 
tions. 


ALGEBRAIC  EXPRESSIONS.  15 

ALGEBRAIC  EXPRESSIONS. 

21.  An  Algebraic  Expression  is  an  algebraic  symbol  or 
combination  of  algebraic  symbols,  representing  some  quan- 
tity. 

For  example,  bx^y  —  Qah  +  7  Vax. 

In  algebra  the  terms  "  number,"  "  quantity,"  and  "  algebraic 
expression  "  represent  aspects  of  the  same  thing,  and  may  be 
used  interchangeably  to  express  these  aspects  as  occasion  may 
require. 

22.  A  Term  is  a  part  of  an  algebraic  expression  contained 
between  a  plus  or  minus  sign  and  the  next  plus  or  minus  sign 
(neither  of  the  two  plus  or  minus  signs  being  inside  a  paren- 
thesis). 

Ex.  1.  dx'y  -  6ab  +  7  y'ax. 

This  algebraic  expression  contains  three  terms — viz.  bx^y^ 
—  Qab,  and  7  vax.  - 

Ex.2.5x  +  a-^b-\-c. 

This  expression  also  contains  three  terms,  5x,  a-^h,  and  c. 

Ex.  3.  7ax^  +  5(a  +  6)  -  c^ 

Since  the  parenthesis,  (a  +  6),  is  treated  as  a  single  quan- 
tity, three  terms  occur  in  this  expression  also — viz.  7a^j 
5(a  +  6),  -c\ 

23.  A  Monomial  is  an  algebraic  expression  of  only  one 
term ;   as,  5x'^y,  or  c. 

24.  A  Polynomial  is  an  algebraic  expression  containing 
more  than  one  term. 

Ex.  Sab-c  +  2x  +  by\ 
.  A  monomial  is  sometimes  called  a  simple  expression^  and  a 
ipolynomial  a  compound  expression. 

25.  A  Binomial  is  an  algebraic  expression  of  two  terms ; 
as,  2a  -  36. 


16  ALGEBRA. 

A  Trinomial  is  an  algebraic  expression  of  three  terms ;  as, 
2a  -  36  +  5c. 

2G.  Positive  and  Negative  Terms.  ;  Terms  preceded  by 
the  plus  sign  are  called  positive,  and  those  preceded  by  the 
minus  sign  are  called  negative.  If  no  sign  is  written  before  a 
term  (as  at  the  beginning  of  an  expression),  the  plus  sign  is 
understood. 

In  the  expression  dx^y  —  Qab  +  7  the  positive  terms  are  5x^y 
and  7.    The  negative  term  is  —  Qab. 

27J[^imilar  and  Dissimilar  Terms.)  Similar  (or  like)  terms 
are  those  which  have  the  same  literal  factors  with  the  same 
exponents  (the  coefl&cients  and  signs  of  similar  terms  may 
be  unlike). 

Ex.  7ab^  —  hah^  are  similar  terms. 

Dissimilar  (or  unlike)  terms  are  unlike  either  in  their  let- 
ters or  the  exponents  of  the  letters. 
Ex.  ba})^^  5ab^  are  dissimilar  terms. 

28.  Reading  Algebraic  Expressions.  It  is  important  at 
this  point  that  the  student  acquire  the  power  of  read- 
ing algebraic  expressions— i.  e.  of  converting  algebraic  sym- 
bols into  ordinary  language,  and  conversely  of  converting 
language  expressing  relations  of  quantity  into  algebraic 
symbols. 

Ex.  1.    Read    the    algebraic    expression,   5a'  —  Q(x  +  y) 

Expressed  in  ordinary  language,  this  expression  becomes, 
"  five  times  a  cube,  minus  six  times  the  quantity  x  plus  y, 
plus  seven  times  c  square." 

Ex.  2.  Express  in  algebraic  symbols  the  following:  six 
times  a  cube,  plus  five  times  b  square,  minus  three  timee 
the  sum  of  a  and  b. 

We  obtain  6a' 4  56' -  3(a -}- 6). 


ALGEBRAIC  EXPRESSIONS.  17 

EXERCISE  1. 

Read  and  copy  the  following  algebraic  expressions: 

1.  9a;  -  ny\  7.  bx  -  2x{\  +  2.x). 

2.  2ah  +  Wc.  8.  Is^y  -  ^x^  +  ^xy\ 

4.  ^(x-2y)-z\  X         Zy  ^  ^' 

5.  2  +  3(x-4).  5-3(r>4-l)      2^ y_^ 

6.  34a;-l)-ar'.  *  5(a;^  +  l)-3^  y"       3a;*' 

11.  (^  -  2a)  (7a;  +  11a)  <  ^^TtS* 

(a'— x'  +  l)' 

12.  •  ["^^  ^^'  ^  5K  -  3^)-  (2/^  -  2n-z). 

Express  in  algebraic  sj^mbols — 

13.  Five  times  a  plus  seven  times  h. 

14.  Six  a  square  equals  twice  the  quantity  a  minus  h. 

15.  Four  times  the  quantity  a  square  minus  nine  h 
is  less  than  the  square  of  the  quantity  seven  a  plus  h 
cube. 

16.  The  product  of  x  minus  ten  y  square,  and  x  cube  plus 
y  times  2,  equals  two  a  times  x  to  the  eighth  power. 

17.  The  quantity  nine  x  plus  two  y  divided  by  three  z,  is 
equal  to  nine  x,  plus  two  y  divided  by  three  z  cube. 

18.  Five  a  cube,  plus  six  ft  square  over  the  square  of  the 
quantity  x  minus  two  y  cube,  is  greater  than  five  times  the 
quantity  a  cube  plus  h  square,  over  the  cube  of  the  quantity 
X  plus  two  y  to  the  fourth  power. 

19.  Four  X  minus  a  fraction  whose  numerator  is  x  plus  y 
square,  and  whose  denominator  is  the  square  of  the  quantity 
X  plus  2/,  equals  one,  minus  x  square  over  y. 

29.  The  Numerical  Value  of  an  algebraic  expression  is 
obtained  by  substituting  for  each  letter  in  the  expression  the 
2 


18  ALGEBRA. 

value  which  it  represents,  and  performing  the  operations  in- 
dicated. 

Thus,  if  a  =  1,6  =  2,  c  =  3. 

Ex.  1.  Find  the  numerical  value  of  lah  —  c^ 

7a6  -  c'  =  7  X  1  X  2  -  3^ 
=  14-9 
=   5,  Ans. 

96 
Ex.  2.  Find  numerical  value  of     -  +  bah''  -  7(a'  +  26)'  +  2>e. 

c 

We  obtain  =  ^^^  +  5  X  1  X  2'^  -  7(1'  +  2  X  2)M  3  X  3^ 
o 

=  6  -f  20  - 175  +  27 

=  - 122,  Ans. 

EXERCISE   2. 

Find  the  numerical  value  of  each  of  the  following  when 


=  5,  6  =  3; 

,  c  =  l,  x  =  Q: 

1.  2a. 

11.  a  +  2c. 

21.  3  +  2(a;-a). 

2.  3x. 

12.  36 -a:. 

22.  5x  -  3(26  +  c). 

3.  ax. 

13.  x'  —  ax. 

23.  2(a;' -  a')  +  3ac. 

4.  3ac. 

14.  2a; -46c. 

24.  7a:(5a  -  4x)  -  ca:^ 

6.  h\ 

15.  a'-bx\ 

25.  3a;(x  -  3)' -  9a;. 

6.  d'c. 

16.  3(a  -h  c). 

26.  (a;  -  1)  (a;  -  3)  +  a;(a;  -  a). 

7.  ex. 

17.  <a-6). 

27.  3(2a;-5c)-a(26-^-3a;). 

8.  3a6^ 

18.  2b(x-c'). 

28.  a(3a;-a6)'  +  a;(a'6'-a;'). 

9.  a'b. 

19.  4(a  — 3c)'. 

29.  (56  +  a;)(a;-6+a-5c'). 

10.  lahc\ 

20.  2x(2a-Sby. 

30.  3a;(a;  -  a)  (a;^  -  a6"^  +  2ac'). 

:u.  a/^6*''c-6Va;  + 

cV-(2a;  +  a-6)l 

If  a  =  ■^,  6  =  1,  a;  =  2,  2/  "=  f  ^  fi^d  values  of — 

32.  6a.  34.  a6a:.  36.  Sab\  38.  2a;  +  53/. 

33.  by.  35.  a-V.  37.  x  -  26.  39.  6rt6  -  b^y. 


ALGEBRAIC  EXPRESSIONS.  19 

40.  6(102/ -36).  43.  ^a  +  a(Zx  -  Khj). 

41.  3x(4a  +  36).  44.  bx  -  Z{by  -  ah). 

42.  ax  +  bx{Zb  —  y).  45.  bx(hy  —  a^)  —  hx. 

46.  6(a +  6)^^  + 10(7/ -a)^ 

If  a  =  4,  6=1,  c  =  0,  a;  =  1,  2/  =  9,  find  values  of— 

47.  l/^  49.  l/ox^  61.  ^VWyl 

48.  I^2a^.  50.  v'a^  52.  361/66^. 

53.  1^0^=^.  •        59.  3a  -  l/46M=~By. 

54.  xVW^^aS.  60.  5a;  +  a:l/95^^Ta'. 

55.  ah\^y'  —  d'  —  x.  61.  3:c l/a%^"5^- 6 V. 

56.  3ca;T/^T6^':  62.  (VTc  +  4^/)  (l/lla +  5x). 

57.  a6v'2a=*-26ar'.  .       63.  l/a6'  +  6c'  +  ex'  +  xa\ 

58.  a  +  l/p^.  64.  (a  -  26)  l/^^li?^ 

Find  the  numerical  value  of — 

65.  {x  +  3)  (2a;  -l)-bx  when  x  =  2. 

66.  3(2a;  -  1)  +  6a;  +  7  when  a;  =  3. 

67.  3(6a:  +  1)^  +  2(x  -  3)  -  5t  when  a;  =  4. 

68.  8a;(2a;  +  1)  -  3(a;  +  1^  when  a;  =  1. 

69.  3a;(2a;  +  l)'^  +  5a;  -  (a;  +  2)'  when  a;  =  2. 

70.  2a;(a;  +  3)  (a;  +  2)  -  (x  +  4)'  when  a;  =  5. 

71.  8a;'  -  5a;(a;  +  2f  -  Q>x^  when  a;  =  3. 

72.  3a;(a;  +  1)*  -  6a;  -  7(a;  +  2)  when  a;  =  4. 

73.  Ta;  +  5(4a;  +  1)  -  3a;(a;  +  2)'  when  a;  =  1. 

74.  ^7?{x  +  \y  -  (a;  +  2)^^  (x  -  If  when  x  =  2. 

75.  (3a;  +  yf  -  2(a;  -  2y)  (a;  +  2y)  +  5(2a;  -  Syf  when  x  =  ^ 
t/  =  2. 


CHAPTER    II. 

METHODS  OF  USING  ALGEBRAIC  SYMBOLS. 

30.  Second  Source  of  New  Power  in  Algebra.  The 
second  general  source  of  new  power  in  algebra  lies  in  cer- 
tain standard  ways  or  methods  in  which  the  symbols  of 
algebra  are  used.  These  ways  are  termed  the  Laws  of  Al- 
gebra. There  are  two  divisions  of  the  primary  laws  of 
algebra : 

I.  Laws  for  +  and  —  Signs. 

II.  Law^s  for  Grouping  and  Arrangement  of  Symbols 
OF  Quantity. 

I.  Laws  for  +  and  —  Signs. 

81.  First  Law  for  +  and  —  taken  together.  As  was 
explained  in  Arts.  8,  9,  10,  the  signs  +  and  —  are  employed 
for  two  purposes — first,  to  express  positive  and  negative 
quantity ;  and  second,  to  indicate  the  operations  of  addition 
and  subtraction.  We  are  able  to  put  these  signs  to  this  double 
use  because,  as  used  in  both  of  these  ways,  the  signs  are  gov- 
erned by  the  same  laws. 

Thus,  if  the  distance  to  the  right  of  0  be  regarded  as  posi- 
tive, and  therefore  the  distance  to  the  left  of  0  as  negative, 

—7—6—5—4  —8  —2  —1         +1  +2  +3  +4  -f  5  +6  +7  +8 

XU I I \ \ \ I         I        I I \ \ \ I I \ I E' 

^  K       B         0       A  F  ^ 

and  a  person  walk  from  0  toward  E  a  distance  of  5  miles 
(to  F)^  and  then  walk  back  toward  W  a  distance  of  3  miles 
(to  A)^  the  distance  travelled  by  him  may  be  expressed  as 

20 


METHODS  OF  USING  ALGEBRAIC  SYMBOLS.         21 

the  sum  of  a  positive  quantity  and  a  negative  quantity; 
that  is, 

(positive  distance  OF)  +  (negative  distance  FA)^ 
or,  +5  +  (-3)  =  5-3  =  2. 

The  position  arrived  at  raay  also  be  determined  in  another 
way — viz.  by  deducting  (that  is,  using  the  operation  of  sub- 
traction) 3  miles  from  5  miles.     We  obtain 

5-(4-3)=5-3  =  2. 

Hence,  we  see  that  adding  negative  quantity  is  the  same  in 
effect  as  subtracting  positive  quantity;  therefore,  in  the  ex- 
pression 

5-3 

the  minus  sign  used  may  be  considered  either  a  sign  of  the 
quality  of  3,  or  as  a  sign  of  operation  to  be  performed  on  3. 

Hence,  we  are  able  to  use  the  signs  +  and  —  to  cover  two 
meanings,  as  stated  above. 

Whichever  of  these  two  meanings  be  assigned,  we  see  that 
4-  (-  3)  =  -  3 ;  also,  -  (+  3)  =  -  3.     Hence, 

Law  1.  The  signs  +  and  —  applied  in  succession  to  a  quantity 
are  equivalent  to  the  single  sign  — . 

Or  in  symbols, 

+  (—  a)  =  —  a ;  —(+  a)  =  —  a. 

32.  Second  Law  for  +  and  —  taken  together.  Again, 
if  in  the  above  illustration  a  person  walk  in  the  negative 
direction  from  0—i.  e.  toward  W—a  distance  of  4  miles  to 
K,  and  then  reverse  his  direction  and  go  2  miles,  he  will  be 
at  jB;  or  the  distance  travelled  is  expressed  as 

-4-(-2)  =  -4+2  =  -2; 

that  is,  the  two  minus  signs  in  —  (—  2)  taken  together  give  +. 

So  also  we  see  that  deducting  a  certain  sum  from  a  man's 

debts  is  the  same  in  effect  as  adding  this  sum  to  his  assets ; 


22  ALGEBRA. 

or,  in  general,  that  a  double  reversal  of  the  quality  of  any 
quantity  gives  the  original  quality.     Hence, 

Law  2.  The  sign  —  applied  twice  to  a  given  positive  quantity 
gives  a  -\-  result. 

Or  in  symbols,  —  (  —  a)  =  +  a. 
It  is  also  evident  that  +  (+  a)  =  +  a. 

By  these  laws  any  succession  of  +  and  —  signs  applied  to 
a  quantity  can  be  at  once  reduced  to  a  single  +  or  —  sign. 

Ex.  -[+(-a)]  =  -[-a]  =  +  a. 

These  laws  therefore  enable  us  to  use  negative  quantity 
with  as  great  freedom  as  we  use  positive  quantity,  and  hence 
are  an  important  source  of  power.  They  also  open  the  way 
to  a  free  use  of  the  second  group  of  the  laws  of  algebra. 

n.  Laws  of  Arrangement  and  Grouping. 

33.  Formal  Statement  of  Laws  of  Arrangement.  The 
laws  which  govern  the  arrangement  and  grouping  of  the 
symbols  for  quantity  in  algebra  are — 

A.  The  Commutative  Law. 

1.  For  Addition,  a  +  b  =  b  -\-  a. 

2.  For  Midtiplication,  ah  =  ha. 

3.  For  Division,  a^hXc  =  aXc^b. 

B.  The  Associative  Law. 

1.  For  Addition,  a  +  6  +  c  =  a  +  (6  +  c)  =  (a  +  6)  +  c. 

2.  For  Mxdtiplication,  ahc  =  a(hc)  =  (ah)c. 

C.  The  Distributive  Law. 

1.  For  Multiplication,  a(h  -\-  c)  —  ah  -+•  ac.     Hence,  in- 

versely, ah  +  ac^  a(h  +  c). 

__,^.  ..       6  +  c6,c 

2.  For  Division,  =  -  H 

a         a      a 

34.  Meaning  of  Commutative  Law  for  Addition.  The 
meaning  of  these  laws  is  best  shown  by  examples.     If  it  is 


METHODS  OF  USING  ALGEBRAIC  SYMBOLS  23 

required  to  combine  a  group  of  7  objects  and  another  group 
of  5  objects  into  a  single  group,  we  may  either  count  the  7 
objects  first,  and  then  count  on  the  5  objects  afterward,  or  the 
5  objects  first  and  the  7  afterward ;  that  is,  groups  of  objects 
may  be  counted  together  into  a  single  group  in  any  order  we 
please.  The  algebraic  symbols  representing  different  groups 
in  like  manner  may  be  arranged  in  any  order  we  please;  that 
is,  briefly, 

a-\rh  =  h  -\-  a. 

An  example  of  the  advantage  in  this  quality  of  algebraic 
symbols  is  that  similar  terms  in  an  expression  may,  by  rear- 
ranging the  terms,  be  brought  together,  and  then  counted  into 
a  single  term  by  the  use  of  the  Distributive  Law  for  Multipli- 
cation (inverse  form). 
Thus,  the  terms  of  the  algebraic  expression, 

la^h  -  bxf  +  ^xy"  +  Za'h  +  Axf  -  2a\ 

by  the  use  of  the  Commutative  Law  may  be  arranged  thus, 

la^h  +  ^o?h  -  2a'6  -  bxy^  +  e>xif  +  Axy\ 

By  the  Distributive  Law  for  Multiplication  (inverse  form) 
the  first  three  terms  may  be  combined  into  a  single  term,  and 
the  last  three  into  another  term,  giving 

Wh  +  hxy\ 
Thus,  by  use  of  these  laws  6  terms  are  reduced  to  2  terms. 

The  symbols  used  to  represent  number  in  arithmetic  cannot  be  changed 
about  in  this  manner.  Thus,  the  number  234  cannot  be  written  324.  If, 
however,  we  employ  the  +  sign,  the  symbols  used  for  the  number  may  be 
put  in  a  commutative  form ;  as, 

234  =  200  +  30  +  4 
=    30+    4  +  200 
=  etc. 

The  arithmetical  form,  234,  has  the  advantage  of  greater  brevity  than 
the  algebraic  form,  200  +  30  +  4,  but  the  disadvantage  of  less  flexibility. 

35.  Meaning  of  Commutative  Law  for  Multiplication. 


24 


jiLGEBRA. 


To  illustrate  the  Commutative  Law  for  Multiplication,  we 
recognize  that  if  we  have  15  objects,  the  number  of  the  ob- 
jects is  the  same  whether  they  be  arranged  in  5  rows  of  3 
objects  each  or  3  rows  of  5  objects  each — 


•  •        •        •        • 

•  •        •        •        • 

•  •       •       e       • 


So,  in  general, 


ah  =  ha. 


The  advantages  resulting  from  this  property  of  algebraic 
symbols  are  illustrated  by  the  fact  that  we  are  enabled  by  it 
to  have  a  standard  order  for  the  arrangement  of  the  literal 
factors  in  a  term — viz.  the  alphabetical  order.  Thus,  instead 
of  writing  Ic^xa^  or  lax&.  or  lxac\  we  write  the  literal  factors 
in  the  alphabetical  order,  la&x. 

,  When  the  factors  in  each  term  of  an  expression  are  thus 
arranged,  it  is  much  easier  to  recognize  similar  terms. 

86.  Meaning  of  the  Distributive  Law  for  Multiplioa- 
tion.     Let  it  be  required  to  reduce  the  expression, 

6(a  -  6  +  c)  +  2(a  +  6  -  c)  +  3(a  +  6  +  c), 
to  its  simplest  form. 

Applying  the  Distributive  Law,  the  expression  becomes 

ba  -  56  +  5c  +  2a  +  26  -  2c  +  3rt  -H  36  +  3c. 
Using  the  Commutative  Law,  we  obtitin 

5a  +  2a  +  3a  -  56  +  26  +  3S  f  5c  -  2c  +  3c. 
Hence,  by  the  Distributive  Law, 
10a -f  6c, 


METHODS  OF  USING  ALGEBRAIC  SYMBOLS.         25 

In  other  cases  the  Distributive  Law  enables  us  to  perform 
work,  part  by  part,  which  would  be  difficult  if  not  impossible 
in  the  undivided  form. 

In  general,  therefore,  these  laws  enable  us  to  arrange  and 
group  the  parts  .of  an  algebraic  expression  to  the  best  advan- 
tage according  to  the  work  to  be  done.  They  are,  therefore, 
to  be  considered,  from  one  standpoint,  as  economic  methods 
which  govern  the  use  of  algebraic  symbols. 

It  will  be  a  useful  exercise  for  the  student  to  determine 
which  of  the  fundamental  laws  for  grouping  and  arranging 
algebraic  symbols  are  used  in  the  following  illustrative  ex- 
amples : 

Ex.1.   6(x  +  y)  +  S(x-y  +  z)-\-2(x  +  2y-z), 

=   6x  +  Qy  +  Sx-Sy  +  Sz  -\-  2x  +  4y  —  2z 
=   6x  +  Sx  +  2x  +  Qy-Sy  +  4ty  +  Sz-2z 

=  llx  +  7y  +  z  . 

^    ^    12a'b'  +  9a'b'  +  6a'b' 
xLiX.  Z. 


Ex.3. 


Bab 

= 

12a:'b'      9n'b' 

Qa'b' 

Sab         Sab 

Sab 

= 

4a'b'  +  Sa'b   +  2ab' 

=ab(4:ab+Sa     +  2b) 

(2x 

+  32/)  (3. 

*  +  42/). 

'  =  2x(^ 

\x+   4y)+Sy(S: 

c+  42/) 

=  Qx' 

+  8xy  +  dxy 

+  12y' 

=  6x' 

+  17xy  +  12y' 

CHAPTER    III. 

ADDITION   AND   SUBTRACTION. 

ADDITION. 

37.  Addition,  in  algebra,  is  the  combination  of  several 
algebraic  expressions,  representing  numbers,  into  a  single 
equivalent  expression. 

38.  Addition  of  Similar  Terms.  If  the  question  be  asked, 
How  many  books  are 

3  books  +  7  books  +  4  books  ? 

the  answer  is,  14  books. 

In  like  manner,  if  the  question  be  asked,  How  many  a^6^'s 
are 

the  answer  is,  14a'6'. 

The  simplification  is  obtained  by  the  use  of  the  Distribu- 
tive Law  for  Multiplication  (Art.  36). 

Thus,  similar  terms  are  added  by  adding  the  coefficients 
of  the  terms  and  setting  the  result  before  the  literal  part 
common  to  the  terms. 

If  some  of  the  similar  terms  are  negative,  the  sum  of  the 
coefficients  is  taken,  respect  being  had  to  their  signs.  The 
sum  thus  taken  is  called  the  algebraic  sum  of  the  coefficients. 

For  example,  add  the  similar  terms 

Sa'x  -  7a'x  -  Qa'x  +  lOa'x  -  a'x. 

The  sum  of  the  plus  coefficients  is  +  18,  the  sum  of  the  nega- 
tive  coefficients  is  — 14,  the  algebraic  sum  of  +18  —  14  is 
-f  4 ;  hence,  the  sum  of  all  the  given  similar  terms  is  +  Aa^x, 
26 


ADDITION.  27 

39.  Addition  of  Dissimilar  Terms.  If  the  terms  to  be 
added  are  dissimilar,  the  addition  of  them  can  be  indicated 
only. 

Thus,  h  added  to  a  gives  a-\-h;  also,  a?  —  Sd^b  added  to 
Sa'  -  b'  gives  a'  -  Sa'b  +  3a'-  61 

Simplifications  are  possible  only  where  there  are  similar  terms. 

40.  General  Method  of  Addition.  The  most  convenient 
general  method  for  addition  is  shown  in  the  following  ex- 
amples : 

Ex.1.  Add  4a;' +  8a;  4- 2,  3a;' -  4a;  -  3,  -2x'-x-6. 

Arranging  similar  terms  in  the  same  column,  and  adding 
each  column  separately,  we  obtain 

4a;'  +  3x  +  2 
8a;'-4a;-3  '     . 

-2:g'-   x-5 
5x''-2x~6,Sum. 

Ex.  2.  Add  2a'  -  5a'6  +  4a6'  +  a'b\  4a'6  +  2a'  -  ah'  -  Bab\ 
a'b-a'  +  2ab\ 

Proceeding  as  in  Ex.  1, 

2a'-5a'6  +  4a6'  +  a'6' 

2a' +  4a'6  -  3a6'  -ab' 

-a'+    a'b  +  2ab' 

3a'  +  Sab' +  a'b' -  ab\  Sum, 

In  the  second  column  the  algebraic  sum  of  the  coefficients  is 
—  5  +  4  +  1,  which  =  0 ;  and  as  zero  times  a  number  is  zero, 
the  sum  of  the  second  column  is  zero,  which  need  not  be  set 
down  in  the  result. 

Hence,  the  general  process  for  addition  may  be  stated  as 
follows : 

Arrange  the  terms  to  be  added  in  columns,  similar  terms  in  the 


28  ALGEBRA. 

same  column;  in  each  column  take  the  sum  of  the  +  coefficients, 
and  also  the  sum  of  the  —  coefficients; 

Subtract  the  less  sum  from  the  greater,  prefix  the  sign  of  the 
greater,  and  annex  the  common  letters  with  their  exponents. 

41.  Collecting  Terms.  It  is  often  required  to  add  together 
the  similar  terms  which  occur  in  a  single  polynomial.  This 
is  called  collecting  terms. 

Ex.  Simplify  2x  +  7ab  +  5-  Sab  +  2ab  +  Sx 

Collecting  terms,  we  obtain  5a;  -f  5  +  6a6. 


EXERCISE  3. 

Find  the  sum  of- 

— 

1. 

2. 

3. 

4. 

5. 

-11 

4 

8a; 

—  X 

-7a; 

6 

- 

-10 

-6a; 

-Sx 

12a; 

6. 

7. 

8. 

9. 

10. 

2a 

- 

-x' 

7xy 

a'b 

7xy 

5a 

Sx' 

-lOxy 

ba'b 

-  10a;y 

- 

-12a 

5x' 

2xy 

-Sa'b 

a^if 

11. 

Sax,  - 

-  2a,x, 

,  bax, 

ax, 

-  Sax. 

12.  5x\  12a:^  -  10:c^  x\  -  IQa^,  Sx\  -x\ 

13.  7a*-'6^  -  12a'b\  -a'b\  -4a'b\  5a'b\  Qa'b\ 


14. 

15. 

16. 

Sx-2y 

5a:^+    7 

a^  —   ax-\-  Aa^ 

2x  +  Sy 

a;^-10 

3a'^  +  2aa;  — 5a;' 

X-   y 

-7a;^+    1 

—  a^—   ax—  x^ 

17.  a  —  2b,  Sa  +. 46,  a  +  5b,—5a  —  b,a  —  5b. 

18.  Sx"  +  y\  2x^  -  1y\  -  4a:^  -  5y\  x'  +  Sy",  -  Sy\ 

19.  Sa^  -  5bif,  2ax^  +  46?/,  2by^  -  4aa;',  63/  -  aar'. 


SUBTRACTION.  29 

20.  a^-xy  +  Sy\2x'  +  2xy-2y\x'-\-y\Sx'-xy. 

21.  mn  —  3n^  +  m^,  m^  +  2n^  —  Zmn,  m^  —  n\  mn  —  2m^ 

22.  x'  +  y'  "  2s^  3a;'  -y'+  2z\  ^  -  2x^  x'  -  z\ 

23.  2x'  -  xy,  Zxy  -  by\  Sif  -  ^x\  x'  +  2y'  -  2xy. 

24.  7x-^y-i-bz  —  lOxy,  2y  —  Sz  +  l^xy  —  4xz,  5z  —  Qx  —  4xzy 
+  2xy,  —i^y  +  9z-{-7x  —  xz, '  21xz  —  16z  +  a:  —  5xy. 

25.  x^  +  3xV  -I-  oxy'  +  2/',  ^'^  -  Sx'y  +  3a:2/'  -  2/',  2x'y  -  2a;2/' 
+  2/',  x'  -    y\  x'y  -  Ax"  -  xy'  -  2/^  y'  +  a;'  -  x^  4~  ^^y. 

Collect  similar  terms  in  the  following : 

26.  2a;  -  32/  -  5a:  +  4z  +  42/  +  2  —  2?/  —  a;  -  3z  +  2a;  —  Sy. 

27.  Sxy  —  5ax  +  Sy'  —  2xy  —  3a;'  +  4aa;  —  2y'  +  3aa;  —  2xy. 

28.  x  —  Sy  +  2z  +  22/  —  2x'—  z—  3a;  -  42  -  2a;  +  ;?  +  2a;. 

29.  2a;  -  1  +  52/  -  2  +  3a;  +  2  +  32/  -  3  -  2a;  +  1  -  a;  -  32/. 
80.  3a'6  -  2a'c  +  3a'  -  5a'b  -a'-  Sd'c  +  a'6  +  6a'c  -  2a'. 

31.  5a;^  -  3a;  +  4  -  2a;'  -  6x'  +  4a;  -  7  -  a;'  +  a;^  +  3a;'  -  a; 
+  5  +  3x'  -  6a;  -  a;'  +  4a;  -  2a;'  +  2x. 

32.  2a;"-5a;"'  +  3a;'-a;"  — 7a;  +  3a;'-3  4-  2a;"*  — 5a;'4-5  +  3a;"* 

SUBTRACTION. 

42.  Subtraction,  in  arithmetic,  is  the  process  of  finding 
the  difference  hctween  two  numbers,  and  subtraction  in  alge- 
bra includes  this  work.  But  inasmuch  as  negative  quantity 
is  dealt  with  in  algebra  as  well  as  positive  quantity,  the  word 
difference  takes  a  broader  meaning,  and  we  need  a  broader 
definition  of  subtraction  which  will  cover  both  positive  and 
negative  quantity. 

If  the  quantity  to  be  subtracted  be  named  the  Subtra- 
hend, and  the  quantity  from  which  the  subtrahend  is  taken 
be  named  the  Minuend,  and  the  result  obtained  be  named 
the  Difference,  it  is  evident  that  for  both  positive  and  nega- 


30  ALGEBRA. 

live  quantity,  the  Difference  added  to  the  Subtrahend  will 
give  the  Minuend.     Hence, 

Subtraction,  in  algebra,  is  the  process  of  finding  a  quan- 
tity which,  added  to  a  given  quantity  (the  subtrahend),  will 
produce  another  given  quantity  (the  minuend). 

Thus,  if  we  subtract  Sab  from  10a6,  we  obtain  lab,  for  lab 
added  to  Sab  (subtrahend)  gives  lOab  (minuend). 

43.  Signs  in  Subtraction.  From  Art.  31  it  is  clear  that 
subtracting  a  positive  quantity  is  the  same  as  adding  a  nega- 
tive quantity  of  the  same  absolute  magnitude ;  and  from  Art. 
32,  that  subtracting  a  negative  quantity  is  the  same  as  adding 
a  positive  quantity  of  the  same  absolute  magnitude. 

Hence,  in  subtraction,  the  most  convenient  way  to  govern 
the  signs  is  to  change  (mentally)  the  signs  of  the 
'.^      terms  in  the  subtrahend. 

-rr  Thus,  to  subtract  46  from  76,  we  change  46  men- 

tally to  —46,  and   add   +lb  and  —46,  and   obtain 
the  result  36. 

Again,  to  subtract  —  26  from  76,  we  mentally  change  the 
sign  of  —  26,  and  add  the  result  +  26  to  +  76,  and 
_J^      obtain  +96. 

— ^  (Some    concrete   illustration   of   the   reason   for 

changing  the  sign  of  a  negative  term  of  the  sub- 
trahend to  plus  in  the  process  of  subtraction  should  be  fre- 
quently recalled  by  the  student.  For  example,  subtracting  a 
$10  debt  from  a  man's  possessions  is  the  same  in  effect  as 
adding  $10  to  them.) 

44.  General  Method  for  Subtraction.  Accordingly,  the 
most  convenient  general  method  in  subtraction  is  to — 

Place  the  terms  of  the  subtrahend  under  the  terms  of  the  minu- 
end, similar  terms  in  the  same  column. 

Change  the  signs  of  the  terms  in  the  subtrahend  mentally;  pro- 
ceed as  in  addition. 


SVBTEACTlO^r.  31 

Ex.  1.   From  5a;'' -2x'  +  x~Z  subtract  2x^-Sx'-x  +  2. 

We  obtain  5x'  —  2x'+x  —  B 

27^ -Sx"-   x  +  2  0 

3x'+    x'  +  2x-5,Diff'erence, 

since  the  coefficient  of  ^  is  5  —  2,  or  3,  of  a;Ms  —  2  +  3,  or 
1,  etc. 

Ex.  2.  Subtract  2a*  -  Sa'b  -  6a'b'  -  2ab'  +  26*  from  a*  +  ba'b 
-6a'b'-Sab\ 

We  obtain     a*  +  5a'b  -  6d'b'  -  Sab^ 

2a*  -  3a^6  -  Qa'b'  -  2a6'''  +  26* 


a*  +  8a^6  -   a6'-26* 


The  coefficient  of  a^6*  is  —  6  +  6,  or  0.     The  coefficient  of  6* 


is  0  -  2,  or  -  2. 


EXERCISE  4 

1.        2.        3: 

From  7a6  bx  x 

Take   3a6  9a:  2x 


4. 

5. 

6. 

5x 

-St^ 

-Ixy 

-Sx 

-4x^ 

Zxy 

7.  8.  9.  10. 

From3a;'-4a:  3a;-9  2x^-5  5a;'  +  4a;-3 

Take   2x^4-   x  bx  +  1  -ar'  +  2         -7?-Zx  +  b 


11.  From  3a +  26- 3c- <^  take  2a-26  +  c-2d 

12.  From  7  -  3a;  +  231?  take  15  —  4a;  -  bx\ 

13.  From  a;^  -  2/'  -  2^  +  8  take  2a;'  +  2/'  -  2^'  +  10. 

14.  From  bxy  —  3a;z  +  byz  +  a;'  take  4a:z  —  2xy  —  o^. 

15.  From  2  -  a;  +  a;' +  a:*  take  3  +  a;  -  a;' -  a;' -  2a;*._ 
IB.  Subtract  lOo^y  +  ZxY  -  13a;?/'  from  T^y  -  a;^/'  +  2a^y'. 

17.  Subtract  3  —  2a6  +  3ac  —  Acd  from  5  —  ac  +  Sec?  —  5a(i. 

18.  Subtract  1  +  a;  -  a;' +  a;' -  a;*  from  2  -  a;  -  a;' -  a;' +  a;^. 

19.  Subtract  a  +  26  —  3c  +  4cZ  from  m  +  26-fd  —  x-fa. 


32  ALGEBRA. 

20.  Subtract  Sx' -  2a:'^  +  5a; -  7  from  Sa^  +  2x'-x-7. 

21.  Subtract  —  x'  -  2x'  +  x'  +  5  from  y^  -  x'  +  x''  —  2x 
+  5.  • 

22.  Subtract  Sx"*  —  3a;"  +  a;  —  3  from  x""  +  x""  —  x""  +  x —  1. 

IfA  =  x'-Sx';i-l,B  =  2x'  -5x-S,C=Bx^  +  x'  +  Bx,  find 
the  values  of — 

23.  ^  +  J5  +  C.  25.  A-hB-0. 

24.  B-A  +  0.  26.  A  -B+0. 

USE   OP  PARENTHESIS. 

45-    I.  Removal  of  Parenthesis.     Addition  and  subtrac- 
tion may  be  indicated  briefly  by  the  use  of  the  parenthesis. 
Thus,  the  expression 

2a  +  36  -  5c  +  (3a  -  26  +  3c) 

indicates  that  3a  —  26  +  3c  is  to  be  added  to  2a  +  36  —  5c. 

The  process  of  addition  thus  indicated  by  the  parenthesis 
may  be  performed  in  the  usual  way  by  placing  similar  terms 
in  the  same  column,  etc.  But  expressions  like  the  above 
occur  so  frequently  in  algebra  that  it  is  found  more  conve- 
nient to  simplify  them  simply  by  setting  down  the  terms  to 
be  added  in  succession  (omitting  the  parenthesis)  and  col- 
lecting similar  terms. 

In  accordance  with  this  method  we  obtain, 

2a  +  36  -  5c  +  3a  -  26  +  3c 
=  5a  +  6-2c. 

Similarly,  the  expression 

2a  +  36  -  5c  -  (3a  -  26  +  3c) 

indicates  that  3a  —  26  +  3c  is  to  be  subtracted  from  2a  +  36 
-5c. 

The  most  convenient  way  of  making  the  subtraction  is  to 


USE  OF  PARENTHESIS.  33 

change  the  signs  of  terms  of  the  subtrahend  (dropping  the 
parenthesis  which  contains  them)  and  to  collect  terms. 
Accordingly  we  obtain 

2a  +  36  -  5c  -  3a  +  26  -  3c 

=  —  a  +  56  -  8c. 

Addition  or  subtraction  performed  in  this  way  is  called 
removing  a  'parenthesis.  The  special  rule  to  be  observed  in 
removing  a  parenthesis  is  that — 

When  a  parenthesis  preceded  hy  a  -{-  sign  is  removed,  the  signs 
of  the  terms  inclosed  by  the  parenthesis  remain  unchanged.    But — 

When  a  parenthesis  preceded  by  a  minus  sign  is  removed,  the 
signs  of  the  terms  inclosed  by  the  parenthesis  are  changed,  the  + 
signs  to  — ,  and  the  —  signs  to  +. 

46.  The  Sign  of  the  First  Term  within  a  Parenthesis 
is  usually  +  understood,  it  being  the  custom  to  put  a  plus 
term  first  in  an  algebraic  expression  if  possible.  Owing  to 
the  absence  of  this  +  sign,  the  beginner  frequently  makes  the 
mistake  of  using  the  sign  of  the  parenthesis  as  the  sign  of 
the  first  term  within  it.  This  error  may  be  obviated  at  first 
by  writing  out  the  sign  of  the  first  term  in  the  parenthesis  in 
full,  till  the  fact  of  its  existence  is  firmly  realized. 

Thus,  5a  -  (+  3a  ~  6) 

plainly  reduces  to  5a  —  3a  +  6,  the  —  3a  being  obtained  by 
changing  the  +  sign  before  3a  to  — .  This  is  equally  true 
when  the  +  sign  is  understood,  cs  in  5a  — (3a  — 6) 

=  5a  -  3a  +  6  =  2a  +  6. 

In  both  cases  the  minus  sign  before  the  parenthesis  belongs 
to  the  parenthesis,  indicates  subtraction,  and  disappears  with 
the  parenthesis. 

47.  Parenthesis  within  Parenthesis.  Using  the  paren- 
thesis as  a  general  name  for  the  signs  of  aggregation,  as 
brace,  bracket,  vinculum,  it  is  evident  that  several  parenthe- 


34  ALGEBRA. 

ses  may  occur  one  within   another  in  thie  same  algebraic 
expression.     The  best  general  method  of  removing  several 
parentheses  occurring  thus,  is  as  follows : 
.    Remcyve  the  parentheses  one  at  a  time,  beginning  with  the  inner- 
most ; 

On  removing  a  parenthesis  preceded  by  a  minus  sign,  change  the 
sigiis  of  the  terms  inclosed  by  the  parenthesis  ; 

Collect  the  terms  of  the  result. 

Ex.  Simplify  5a;  —  2/  —  [4a;  —  63/  +  J  —  3a:  +  ^  +  22  -  (2x 

-=bx-   2/-[4a;-62/+ S-3a;  +  2/  +  2z-2a:  +  zn 
=  bx-   2/-[4a;  — 62/        — 3a; +  2/ +  2z  — 2a;  + z] 
—  bx—   y—  4a;  4-  %        -\-Sx  —y  —  2z-{-2x  —  z 
=  6x  +  42/-  3z. 

EXERCISE   5. 

Remove  parentheses  and  collect  similar  terms : 

1.  3a  +  (2a  -  6).  7.  x-  [2a;  +  (x- 1)]. 

2.  2a;  -  (a;  -  1).  8.  5a;  +  (1  -  [2  -  4a;]). 

3.  a;  +  (l-2a;).  9.  2- \1- (S- a)  -  a\. 

4.  3a;  -  (1  +  3a;).  10.  2a;  -  [-  a;  -  (a;  -  1)]. 

5.  a;-(-a;-l).  11.  2y  +  ]- x- (2y -x)\. 

6.  x  +  2y-(2x-y').  12.  a- J-a- (- a- l)i. 

13.  [a;'  -  (x'y  -  z")  -  z']  +  (x'y  -  x'). 

14.  l-Sl-[l-(l--a;)-l]-lJ-a;. 

15.  a;-[-S-(-a;-l)-a;J-l]-l. 

16.  l-12  +  [-3-(-4-5^^-7]S. 

17.  a—\a  +  lb  —  (ia  +  b  +  c  —  a  +  b  +  d)-c]\. 

18.  x-\2x^  +  (Z^  -  3a;  -  [a;  +  x'])  +  [2a; -  (x'  +  a;')]|. 

19.  X*  -  [4ar'  -  l^x'  -  (2a;  +  2)]  +  3a;]  -  [a;*  +  (3a;^  +  2a;*  -  3a; 

-1)].  ^ 


USE  OF  PARENTHESIS.  35 

20.  X  —  x  —  y  —  }  —  a;  —  [—  (a;  —  y)  —  (x  +  2/)  —  «]  —  (« 

-yVs- 

21.  ~l-2x-l-(-2x-l)-2xl-l']-2x. 

22.  x  —  lx-{-(x-y)  —  \x  +  ^  —  x)-2yl-\-y']  —  y-\-  x. 

23.  25a:  -  [12  +  S3x  -  7  -  (-  12a:  -  6  +  15a;)  -  (3  +  2a;){] 

4-  7  -  (3x  +  5)  +  (2a;  -  3)  +  a;  +  8.  ^ 

48.  II.  Insertion  of  Parenthesis.  It  is  plain  that  the 
process  of  removing  a  parenthesis  may  he  reversed ;  that  is, 
that  terms  may  be  inclosed  in  a  parenthesis. 

Inverting  the  statements  of  Art.  45, 

Terms  may  be  inclosed  in  a  parenthesis  preceded  by  the  pltis 
sign,  provided  the  signs  of  the  terms  remain  unchanged; 

Terms  may  he  inclosed  in  a  parenthesis  preceded  by  the  minus 
sign,  provided  the  signs  of  the  terms  be  changed. 

Ex.  a  —  b  +  c  +  d—  e  =  a  —  b  +  (c  +  d  —  e), 

OT,  =  a  —  b  —  (^—  c  —  d  -\-  e). 

EXERCISE   6. 

♦  In  each  of  the  following  insert  a  parenthesis,  inclosing  the 
last  three  terms ;  each  parenthesis  to  be  preceded  by  a  minus 

sign: 


a;^-4. 


1. 

a^- 

-  3a:^  +  3a; 

-1. 

4. 

l-a'-2 

2. 

a- 

-6  + 

c  +  d 

.     6. 

x'  +  4x-: 

3. 

l  +  2a- 

-a'- 

1. 

6. 

a'b'-2cd 

7. 

4a;*- 

-  9a;'  +  12xy  - 

-Ay\ 

8. 

a'- 

-2a  +  l-9- 

-6x-x\ 

9. 

x'- 

-4a;'  +  4a;'  +  4a;-4  — a;^ 

It  is  often  useful  to  collect  the  coeflBlcients  of  a  letter  into  a 
single  coefficient. 


36  ALGEBRA. 

Let  it  be  required  to  collect  the  coefficients  of  x,  2/,  and  z  in 
the  expression, 

Sx  —  4:y-\-5z  —  ax  —  by  —  cz  —  bx-{-ay  +  az. 

The  complete  coefficient  of  a;  is  (3  —  a  —  6) ;  of  1/,  (—  4  —  6 
+  a)  or  —  (4  +  6  —  a)  ;  of  z,  (5  —  c  +  a). 
Hence,  the  same  expression  may  be  written, 

(3  -  a  -  6)a;  -  (4  +  6  -  a)2/  +  (5  -  c  +  a)z. 

In  like  manner  collect  the  coefficients  of  x,  y,  and  z — 

10.  mx  —  ny -{- dz -^  2x -{-  nz  —  4y. 

11.  x  —  y  —  2z  —  ax  -\-by  —  az—bx  —  ay+  cz. 

12.  —7x  +  12y  —  10z  —  2ax-\-Sbz  —  cy  +  2bx  —  Qdy. 

13.  abx  —  bey  —  cdz  +  acx  —  ady  —  acz  —  aby  +  adz. 

14.  by  —  Zacx  —  bcdz  —  Aabx  —  Scdy  +  2cx  —  4z  —  5ax. 

Collect  coefficients  of  x',  x'^,  and  x — 

15.  Sx^  +  X  —  2x^  —  ar'  —  5  +  a'^  —  2ax  —  C7?  —  cx^  —  ex. 

16.  -x^-x-  ax"  +  3?-ax  +  bx'  -  ax'  -  Zbx - 2bx'  -f  Sa. 

17.  aV  -ax -a-  6V  -  26V  +  Sbx  -  aV  -  ex"  +  Sex  -  c. 

EXERCISE   7. 

SPECIAL   REVIEW. 
Add- 

1 .  2t*  -  Sar*  -  Zx^  +  2a;  -  5,  2x^  -  Sx*  -  2x  +  2x^  -  6,  Zx^  +  x*  -  Zx^ 
+  7  -  a:,  and  2  +  Sar'  +  2a;*  -  4x  -  2a;'. 

2.  5a;^0  +  Zx^yz  —  Zxy'^z  —  Zxyz"^,   hxy'^z  —  Zx^yz  —  ^xyz,  and   locyz^ 
—  xyz  —  x^yz  +  xy'^z. 

3.  31/2  -  51/3  +  8,  hV^  -  21/2  -  7,  31/3  -  41/2'  -  2. 

4.  2(a;  +  ^)  -  3(a;  +  0)  +  2(y  +  2;),    4(a;  +  0)  -  3(a;  +  2/)  -  5(y  +  2;), 
and  4(a;  +  y)  —  (a;  +  2)  +  4(2/  +  z). 

Subtract — 

5.  2a6  -  36c  +  d  from  1  -  Zah  -  bo  +  X. 

6.  c  -  (?  +  a;  -  10|2/  from  3a;  -  a  +  c. 


SPECIAL  REVIEW,  27 

7.  19a6  -  c  —  4x  +  Vy  from  \2ah  -  3c  +  c'  -  Vy, 

8.  3  -  2Vx  +  5a;  -  a;^-  a;*  from  21/^  +  a;»  -  1. 

Fintl  value  of — 

9.  3.-C  -{X-  2y  +  2{x  +  1)  (4  -  a;)  -  Vbx  +  1,  when  a;  =  3. 

fd.  6a;2  -  3a:(a;  -h  |)  +  V^x'  -  5a;  +  2,  when  a:  =  1.     When  a;  =  f . 
1  <    c>x-  2(4a;2  -  2a;  -  5)  +  a;(a;  +  I)  (5  -  2a;),   when   a;  =  2.     When 

x^  i. 

Simplify  and  collect — 

12.  3a;  -  {-  2a;  +  [-  4a;  -  (a;  -  2)  -  a;]  -  a;}  -  1. 

13.  9a;  -  {-  8a;  -  [7a;  +  (-  6a;  +  1)  -  5a;]  -  4a;}  -  (3a;  +  1)  -  2a;. 

14.  X'  -  {y^  -  a;2)  -  [(a;^  +  z")  -  {{x"  -  z")  +  [y^  -  z")  -  {x^  +  z^)} 
-  <1. 

Bracket  coeflScients  of  like  powers  of  x — 

15.  a;^  -  ar''  +  2  -  3a;*  -  ax^  +  ax^  -  cx*^  -  lax^  +  3ca;'  -  2ca;*  -  5a;'. 
IG.  1  -  a;  -  a;2  -  a;'  +  2a  -  2aa;  +  lax^  -  2ax^  -  '6bx  +  Zbx''  +  36ar' 

+  ex. 

17.  From  the  sum  of  a^  -  lab  +  36^  and  2a?  -  66^  +  7a'^b\  take  the 
sum  of  4a262  _  3^3  _,.  ^a^  -  6^  and  Sab  -  W  +  a^. 

18.  What  must  be  added  to  a;^  —  a;  +  1  that  the  sum  may  be  a?*  ?    That 
the  sum  may  be  3a;?     15?     0? 

1 9.  What  must  be  subtracted  from  2x^  —  3a;  +  1  that  the  remainder  may 
be  a;'?    a;^  +  10?    7?    a-a;  +  l? 

If  4  =  4a;'  -  Ix'y  +  3a;^2  +    ^3^  C  =  3a;3  _    ^iy  +  ^y%^ 

-B  =  4a:3  -    x^y  -    xy"^  -  ^y^  D  =    x^  -  2xy^  +   y^. 

Find  the  values  of — 

20.  A-  B  +  C-  D.  22.  A  --  (B  +  C)  +  D. 

21.  A-IB-  {D+  C)l  23.  B+  {A-[C-  i)]}. 

24.  B-  {-.A-i-B-(-C)-D]-C}-(C-B). 


CHAPTER    IV. 
MULTIPLICATION. 

49.  Multiplication  is  the  process  of  finding  the  result  of 
taking  one  quantity  as  many  times  as  there  are  units  in 
another  quantity. 

The  Multiplicand  is  the  quantity  to  be  multiplied. 

The  Multiplier  is  the  quantity  showing  how  many  times 
the  multiplicand  is  to  be  taken. 

The  Product  is  the  result  of  the  multiplication.  By  def- 
inition of  "factors"  in  Art.  11  it  is  seen  that  the  multiplier 
and  multiplicand  are  factors  of  the  product. 

Thus,  if  X  is  the  multiplicand  and  y  the  multiplier,  the 
product  is  xy,  and  the  factors  of  xy  are  x  and  y. 

MULTIPLICATION   OP   MONOMIALS. 

50.  Multiplication  of  Coefficients.  To  multiply  4a  by  36, 
we  evidently  take  the  product  of  all  the  factors  of  the  multi- 
plier and  multiplicand,  and  thus  get  4  X  a  X  3  X  6,  or,  rear- 
ranging factors  as  we  are  enabled  to  do  by  the  Commutative 
Law, 

4X3XaX6  =  12ah. 

Hence,  in  multiplying  two  monomials  we  multiply  their 
coefficients  together  to  produce  the  coefficient  of  the  product. 

51.  Multiplication  of  Literal  Factors  or  La^w  of  Expo- 
nents.    To  multiply  d^  by  a' : 

Since  a'  =  a  X  a  X  a 
and  a^^aXa 
.\a'Xa'  =  aXaXaXaXa  =  a\ 


♦  MULTIPLICATION.  39 

This  may  be  expressed  in  the  form 

a^  X  a'  =  a'  +  '  =  a^ 
or,  in  general,  a"*  X  a"  =  a"*  "•■  **, 

where  m  and  n  are  positive  whole  numbers. 

Hence,  in  multiplying  the  literal  factors  of  a  monomial,  we 
add  the  exponents  of  each  letter  that  occurs  in  both  multi- 
plier and  multiplicand. 

Ex.  A.a^hc'  X  U'h'x  =  12a'h'&x. 

52.  La"W  of  Signs.  The  law  of  signs  in  multiplication 
follows  directly  from  the  general  law  of  signs  as  stated  in 
Art.  31. 

To  proceed  by  way  of  illustration: 

(1)  +  $100  taken  5  times  gives  +  $500, 

or,  in  general,   +  quantity  taken  a  +  number  of  times  gives 

a  vf  result. 

(2)  $100  of  debts— that  is,  —  $100,  taken  5  times,  gives 

-  $500, 
or,  in  general,  —  quantity  taken  a  +  number  of  times,  gives 
—  quantity  as  a  result. 

(3)  $100  deducted  5  times,  or  $100  X  —  5,  gives  as  total 

amount  of  deduction  —  $500, 
or,  in  general,  H-  quantity  taken  a  —  number  of  times,  gives 
—  quantity  as  a  result. 

(4)  Deducting  $100  of  debts  5  times  from  a  man's  pos- 

sessions is  the  same  as  adding  $500  to  his  assets ; 
that  is,  -  $100  X  -  5  =  +  $500, 
or,  in  general,  —  quantity  taken  a  —  number  of  times  gives 
+  quantity  as  a  result. 

Thus,  we  see  from  (1)  and  (4)  that 

either  +  X  +,  or  —  X  —  j  gives  -[-, 


40  ALGEBRA.  " 

and  from  (2)  and  (3),  that 

either  —  X  +,  or  +  X  — ,  gives  — ; 
or,  in  brief,  that  in  multiplication 

Like  signs  give  plus,  unlike  signs  give  minus. 

53.  Multiplication  of  Monomials.  Combining  the  results 
of  Arts.  50,  51,  52,  the  process  of  multiplying  one  monomial 
by  another  may  be  expressed  as  follows: 

Multiply  the  coefficients  together  for  a  new  coefficient; 

Annex  the  literal  factors^  adding  the  exponents  of  each  letter  that 
occurs  in  both  multiplier  and  multiplicand  ; 

Determine  the  sign  of  the  result  by  the  rule  that  like  signs  give  +, 
unlike  signs  give  — . 

Ex.  1.   Multiply  da'bx^  by  -  6a6y. 

The  product  is  -  SOa'iV^/l 

Ex.  2.   Multiply  Ba""  + '  by  2a"  "  \ 

Since  n  +  S  and  n  —  1,  added,  give  2n  -f  2, 
the  product  is  10a'"  + ' 

MULTIPLICATION    OP  A  POLYNOMIAL   BY  A 
MONOMIAL. 

64.  Since,  by  the  Distributive  Law,  Art.  33, 

a(b  +  c)  =  ab  -\-  ac, 

it  follows  that  to  multiply  any  polynomial  by  a  monomial 
we  proceed  thus: 

Multiply  each  term  of  the  multiplicand  by  the  multiplier^  and  set 
down  the  results  as  a  new  polynomial. 

Ex.  Multiply  2a'  -  5a'6  +  3a6'^  by  -  Zab\ 
2a'  -  ba'b  +  3a6^ 

-Ub' 

-  6a*6'  +  15a'6'  -  9a'6*,  Product 


MULTIPLICATION.  41 


EXERCISE 

8. 

1. 

2. 

3. 

4.                  5. 

6. 

Multiply 

-5 

-3a 

Zah 

30a;y             4x 

-5x 

By 

4 

7. 

-2        - 
8. 

-5 
9. 

-1               -2x 

=_?? 

10.             11. 

12. 

Multiply 

3ax 

-6x2/' 

7ax 

-  ba'h          Q^ed 

-2x^2/2 

By 

—  Aax 

-"Ixf     - 

-3a2/ 

-4ccf      -Zed' 

-8x2/¥ 

13. 

14. 

15. 

16. 

17. 

Multiply 

Wcdx^ 

-4x« 

5a:Y 

3a;Y-i 

^2„^n-3 

By     - 

-zed 
18. 

3x'* 
19. 

-7xY 

-xy+^ 

-  x"2/" ""  ' 

20. 

21. 

Multiply  2a  +  3a; 

3x- 

-22/ 

40:^2/  ~  ^2/^            ' 

7aa;  —  Aby 

By 

Ux 

-bxy 

2a;2/                 —  3a6a:2/ 

Multiply — 

22.  Sac'  -  3m'n  by  ban.  26.  Sx**  + '  +  Vx**  by  -  4a;. 

23.  m-m'-  3m'  by  -  Im^n.  27.  2x'"  -  5x"2/  by  Zx^y"^. 

24.  8x'2/  —  bxy'  —  y^  by  3x2/.  28.  ax""  —  Iby""  by  x^- 

25.  2x"  —  3x'*  -  '  by  x\  29.  5x' "  ^  —  3x'  -  "  by  4x^ 

30.  2x'^  +  '-3x"  +  '  — x"  +  '  — x"by  5x"-'. 

31.  x"2/  +  3x'*  +  y  -  Ax-  +  y  by  -  2x-  -  y. 

MULTIPLICATION    OF  A  POLYNOMIAL   BY  A 
POLYNOMIAL. 

65.  Arranging  the  Terms  of  a  Polynomial.  The  multi- 
plication of  polynomials  is  greatly  facilitated  by  arranging 
the  terms  in  each  polynomial  according  to  the  powers  of 
some  letter,  the  powers  being  taken  either  in  the  ascending 
or  descending  order. 


42  ALGEBRA. 

In  arranging  the  terms  of  a  polynomial  according  to  the 
ascending  powers  of  a  letter,  the  term  containing  the  lowest 
power  of  the  letter  is  placed  first ;  the  term  containing  the 
next  higher  power  of  this  letter  is  placed  next,  etc. 

Ex.  dx^  -\-  Z  —  X  -\-  x^  —  7x^j  arranged  according  to  the  as- 
cending powers  of  x,  becomes 

S-x  +  5x''~7x^  +  x\ 

In  arranging  the  terms  of  an  expression  according  to  the 
descending  powers  of  a  letter,  the  term  containing  the  high- 
est power  of  the  letter  is  placed  first ;  the  term  containing  the 
next  higher  power  is  placed  next,  etc. 

Ex.  a*  +  6*  —  4a^6^  —  5a^6,  arranged  according  to  the  descend' 
ing  powers  of  a,  becomes 

a'-5a'b-4:a'b'  +  b\ 

When  arranging  two  polynomials  for  purposes  of  multipli- 
cation the  same  letter  should  be  used,  and  the  same  order, 
either  ascending  or  descending,  in  both  polynomials. 

66.  Multiplication  of  Polynomials.  The  terms  of  each 
polynomial  having  been  arranged,  we  proceed  to  multiply 
each  term  of  the  multiplicand  by  each  term  of  the  multi- 
plier, and  take  the  sum  of  the  results.  The  reason  for  this 
is  made  clear  by  taking  two  polynomials,  a  +  b  and  c 
-f  d,  and  forming  their  product  by  use  of  the  Distributive 
Law: 

(a  +  b)(c  +  d)=  a(c  -\-  d) -{■  b(c  +  d),  by  Distributive  Law. 

=  ac  +  ad  -{-  be  -i-  bd,  by  a  second  use  of  this  law. 

We  see  that  a  similar  result  is  obtained,  no  matter  how  many 
terms  occur  in  each  polynomial. 

Therefore,  to  multiply  two  polynomials 

Arrange  the  terms  of  the  multiplier  and  multiplicand  according 
to  the  ascending  or  descending  powers  of  the  same  letter  j 


MULTIPLICATION,  43 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the  mul- 
tiplier ; 

Add  the  partial  products  thus  formed, 

Ex.  1.   Multiply  2x  —  Zy  by  Zx  +  by. 

The  terms  as  given  are  arranged  in  order. 
The  most  convenient  way  of  adding  partial  products  is  to 
set  down  similar  terms  in  columns,  thus : 

2x  -  32/ 
Sx  +  by 
Qx^-   9xy  )    ^      .  . 

+  lQa^y-15.v'  3  products, 

Qx^  +     xy—  Iby"^,  Product. 

Ex.  2.   Multiply  2a;  -  a;'  +  1  -  3a^  by  2a;  +  3  —  a;'. 

Arrange  the  terms  in  both  polynomials  according  to  the  ascending  powers 
of  X.     (Why  is  the  ascending  or'der  chosen  rather  than  the  descending  ?) 

1  +  2X-SX''-      2^ 
S  +  2x-    x^ 


3  +  6x-  9x^  -    3a;3 
+  2x  +  4a;2  -    6ar»  -  2a;* 

-    x'^  -    2x^  +  3a;*  +  a;^ 

3  +  8a;  -  6a;2  -  Uxr^  +    x^  +  x^,  Product. 

Let  the  student  also  multiply  the  two  polynomials  together  with  their 
terms  in  the  order  as  first  given,  and  hence  discover  the  advantage  of 
arranging  the  terms  in  order  before  multiplying. 

Ex.  3.   Multiply  Sab  -  4b'  +  2a'  by  -  2b''  +  3a'  -  5ab. 

Arrange  the  terms  in  each  polynomial  according  to  the  descending  powers 
of  a. 

2a?  +    3a6  -    W 
3a2-    5a6  -    26^ 


6a*  +    9a36  -  l2aW 

-  lQo?b  -  IbaW  +  20a6' 

-    4a262  -    6a6'  +  86* 


6a*  -      a'b  -  Zla'b^  +  l^ab^  H-  86*,  ProdmL 


44  ALGEBRA. 

Ex.4.   Multiply  a' +  &' +  c' +  2a6  — ac-6c  by  a  +  6 +  c. 
Arranging  the  terms  according  to  powers  of  a, 
a^  +  2ab  -  ac  +      b^  -      be  +    c^ 

a  +      b  +     c 

a^  +  2a^b  -  a^c  +    ab"^  -    abc  +  ae^ 

4    a^b  +  2a62  -    abc  +  b^  -  b'^c  +  bc^ 

+  d^e +  labc  -  a&  +  b'^c  -  6c''  +  c^ 

a^  +  3a26  +  Safe^  +  6^  +  c^,  Product. 

Ex.  5.   Multiply  a;''"  +  a:''"  +  a;'"  +  1  by  a;™  4-  1. 
a;3m  +    a:^"*  +     a;*"  +  1 
X"*  +  1 

a-4m  _^  2ar'"'  4-  2a;2'»  +  2af'  +  1,  Prodmt. 

67.  Degree  of  a  Term.  Homogeneous  Expressions.  The 
degree  of  a  term  is  determined  by  the  number  of  literal  fac- 
tors which  the  term  contains ;  hence,  the  degree  of  a  term  is 
also  equal  to  the  sum  of  the  exponents  of  the  literal  factors. 

Ex.  la^h(?  is  a  term  of  the  sixth  degree^  since  the  sum  of 
the  exponents  3  +  1  +  2  =  6. 

A  polynomial  is  said  to  be  homogeneous  when  all  its  terms 
are  of  the  same  degree. 

Ex.  ba^h  —  6'  +  ah"^  is  a  homogeneous  polynomial,  since 
each  of  its  terms  is  of  the  3d  degree. 

58.  Multiplication   of  Homogeneous   Polynomials.      If 

two  monomials  be  multiplied  together,  the  degree  of  the 
product  must  equal  the  sum  of  the  degrees  of  the  multiplier 
and  multiplicand. 

For  instance,  in  Ex.  3.  Art.  56,  the  multiplicand  and  multi- 
plier are  both  homogeneous,  and  each  is  of  the  second  degree, 
and  their  product  is  seen  to  be  homogeneous  and  of  the  fourth 
degree. 

The  fact  that  the  product  of  two  homogeneous  expressions 


MULTIPLICATION.  45 

must  also  be  homogeneous  affords  a  partial  test  of  the  accu- 
racy of  the  work.  For  if,  for  instance,  in  the  above  example, 
a  term  of  the  5th  degree,  such  as  5a^6^,  had  been  obtained  in 
the  product,  it  would  have  been  at  once  evident  that  a  mis- 
take had  been  made  in  the  work.  The  student  should  make 
use  of  this  principle  in  testing  the  results  obtained  in  exam- 
ples 8,  9,  10,  12,  14,  15,  20,  21,  22,  23,  24,  25,  of  the  following 
exercise. 

EXERCISE   9. 

Multiply— 

1.  a;-4by  2a;+l.  b.  1x^ -bf  hy  Ax^ -{-Zf-, 

2.  a;  —  3  by  3a;  +  2.  <>.  bxy  +  6  by  Qxy  —  7. 

8.  2a;  +  5  by  x  -  7.  7.  4a'  -  h'c  by  8aV  +  2ah'c\ 

4.  3x  —  Ay  by  4x  —  Zy.  8.  llx'y  -  Ixf  by  3a;'  +  2y\ 

9.  a'  -  a6  +  6'  by  a  +  h. 

10.  x^  -\-7?y  +  xy^  -{-'jfhyx  —  y. 

11.  4a;'  -  3a;'  +  2a;  -  1  by  2a;  +  1. 

12.  2a;'  —  Zxy  +  2/  by  3a;  -  by. 

13.  a;'  -  3a;'  +  2a;  -  1  by  2a;'  +  a;  -  3. 

14.  Zx'^y  —  Axy^  —  if  by  a;'  —  2xy  —  y^. 

15.  x^  -  3a;'7/  f  3.r?y'  -  2/'  by  2;'  -  2xy  +  if. 
1(>.  4x'  -  3a;'  +  5a;  -  2  by  a;'  +^3a;  -  3. 

17.  .T*  ~- 3a;' +  5  by  a;' —  a;  —  4.^^ 

18.  ^  ~  Zxy  -\-  y^  by  x^  —  Sxy  —  y^, 

19.  a;'-7a;+2by  a;'-7a;-2. 

20.  a'-ah  +  6'  by  a'  +  a6  +  h\ 

21.  4a;'  +  dy'  —  Qxy  by  4a;'  +  9y'  +  ioxy. 
2Z.X*-  7a;'2/'  +  63;*,^  —  y*  by  x'  -  2a;?/'  +  f, 

2S,  x'  -  6aa;'  +  12a'a;  -  Sa'  by  -  a;'  -  4aa;  -  ia\ 
2i,  a'  +  6'  +  a;'  +  2ab  -  aa;  -  6a;  by  a  +  6  +  a;. 
25.  a6  +  cci  +  ac  +  6c?  hy  ab -\-  cd  —  ac  —  bd. 


46  ALGEBRA. 

26.  a:'*  +  2a;''-^ +  3:c~-'-2by  X  — 2. 

27.  a;**  +  '  -  3a:'*  +  4a;" " ^  —  Sa;*^ - '  by  a:"  +  2x'*-\ 

28.  x""-'  —  2x^-'  +  3a:"-'  —  4a;"-^  4-  5a:"  by  2a:'  +  3a:  +  1. 

59.  Multiplication  indicated  by  the  Parenthesis.  Sim- 
plifications. The  parenthesis  is  useful  in  indicating  multi- 
plications or  combinations  of  multipHcations. 

Thus,  (a  —  6  +  2c)'  means  that  a  —  b  +  2c  is  to  be  multi- 
plied by  itself. 

(a  —  6  +  2cy  means  that  a  —  b  +  2c  is  to  be  taken  as  a 
factor  three  times  and  multiplied. 

To  multiply  out  such  a  power  is  termed  to  expand  the 
power. 

Again,  (a  —  b)  (a  ■—  26)  (a  +  b  —  c)  means  that  the  three 
factors,  a  —  6,  a  —  26,  a  +  6  —  c,  are  all  to  be  multiplied 
together. 

Also,  (a  —  2a:)'  —  (a  +  2x)  (a  —  2a:)  means  that  a  +  2a:  is  to 
be  multipHed  by  a  —  2a:,  and  the  product  subtracted  from  the 
product  of  a  —  2a:  by  itself. 

To  simplify  an  expression  in  which  multiplications  are  in- 
dicated in  any  of  the  above  manners,  means  to  perform  the 
operations  indicated  and  to  collect  terms. 

Ex.  Simplify  3(a:  -  2y)  (x  +  2y)  -  (a:  -  2yy. 
3(a:  -  22/)  (a:  +  2i/)  =  3a:' -  122/' 

(a:  —  22/)'  =   a:'  —   4xy  +  4y^ 

Subtracting  the  second  expression  from  the  first,  we  obtain 
•     2a:'  +  4xy  -  16y\ 

EXERCISE  10. 

Simplify  by  removing  parentheses  and  collecting  terms : 

1.  a:'  -  a:(l  +  x).  4.  (a:  -  5)'  -  (a:  +  5)'. 

2.  (x  -  2)  (2a:  +  4).  5.  32:  -  2a:(l  +  x  +  a:'). 

3.  3a:(x  -  2)  -  2a:(a:  -  3).  6.  a:  -  (a:  -  1)  (a:  +  2). 


MULTIPLICATION.  47 

7.  3(a;-3)(x'  +  l)  +  9. 

8.  (a-26)C3a  +  46)~3a'. 

9.  (a  +  26  -  3c)  (a  -  26  +  3c). 
ic.  (a;  -  2/  +  z)'^  -  <x  -  22/  +  2z). 
li.  2:c^-3(a;-l)^  +  (a;-2)l 

12.  Jx^  +  a;(l-a;)(2  +  a:)  +  a;l 

13.  2  -  3(a:  -  2)^  -  2(3  -  2x)  (1  4-  x). 

14.  a'^  —  [x(a  —x)—  a{x  —  a)]  —  x'. 

15.  (x-l)(:8-2)-(a;-2)(a;-3)  +  (a:-3)(a;-4). 

16.  3(:c-2/)'-2K^  +  2/)'-(^-2/)(^  +  2/)S+22/^ 

17.  x(a;  -  2/  -  z)  —  2/(2  —  a;  -  2/)  —  2(2  —  2/  -  x)  -  2/'- 

IH.  3[(a  +  26)a;  +  2my']  —  5[(m  —  c)y  +  6a;]  —  \\{x  —  a)a  +  cy], 

19.  26a6  -  (9a  -  86)  (5a  +  26)  -  (46  -  3a)  (15a  +  46). 

20.  6x'-2x'  +  x'-2iix'  +  x-l)(iSx'-x+l)  +  (Sx'-^2')(2x 

-1). 

If  a  =  3,  x=  —  2,  y  =  —5y  find  the  values  of— 

21.  2aa;.  25.  y''  +  ^x(x-y). 

22.  x^y.  26.  4a;^  —  aa;(4a;  —  y). 

23.  Sx'  +  ay.  21,  3a;  -  5(2a;  +  3). 

24.  xy  -  ax^.  28.  2{x^  ^- y)  -  ay  +  ax", 

29.  2(1  -  2a;)^  -\-(x  +  y)  (a^  +  x). 

30.  (a;  -  1)^  -  3(x  +  1)  (a;  +  2)  -  a;(a;^  -  2)  {y  -  2x). 

31.  3a(a  -  2a;)  -  {a  -  (a  -  1)  (x  +  1)  -  (a  +  xfl  +  5aa; 


CHAPTER   V. 
DIVISION. 

60.  Division  is  the  inverse  of  multiplication,  and  may  be 
defined  as  the  process  of  finding  one  factor  when  the  product 
and  other  factor  are  given. 

The  Dividend  is' the  product  of  the  two  factors,  and  hence 
is  the  quantity  to  be  divided  by  the  given  factor. 

The  Divisor  is  the  given  factor. 

The  Quotient  is  the  required  factor. 

Thus,  if  it  is  required  to  divide  lOxy  by  5a;,  it  is  meant  that 
we  must  find  a  quantity  which,  multiplied  by  6x^  will  pro- 
duce lOxy.  The  factor  5x  is  the  divisor,  lOxy  is  the  dividend, 
and  the  other  factor,  or  required  quotient,  is  evidently  2y, 

61.  General  Principle.  Division  being  the  inverse  of  mul- 
tiplication, the  methods  of  division  are  obtained  by  inverting 
the  processes  used  in  multiplication. 

DIVISION  OF  MONOMIALS. 

62.  Division  of  Coefficients.  In  the  division  of  one  mo- 
nomial by  another  the  coefficient  of  the  quotient  is  obtained 
by  dividing  the  coefficient  of  the  dividend  by  the  coefficient 
of  the  divisor. 

For,  by  multiplication.  Art.  50, 

CoefF.  of  dividend  (i.  e.  of  product)  =  coeffi  of  divisor  factor  X 

coefF.  of  quotient  factor, 

.  • .  Dividing  these  equals  by  coeff.  of  divisor  factor, 

Coeff.  of  dividend 


Coeff.  of  divisor 
4^ 


Coeff.  of  quotient. 


a^,  we  have 


DIVISION.  49 

.  Index  Law  for  Division.     If  a^  is  to  be  divided  by 

a"       aXaXaXaXa 

-T  = =aXaXa  =  ar, 

a'  aXa 

Or,  in  general,  —  —  a*"  ~  " 

where  m  and  n  are  positive  whole  numbers. 

Hence,  in  general,  the  exponent  of  a  literal  factor  in  the 
quotient  is  obtained  by  subtracting  the  exponent  of  this 
letter  in  the  divisor  from  the  exponent  of  the  same  letter  in 
the  dividend. 

64.  Law  of  Signs  in  Division.  This  law  is  obtained  by 
inverting  the  processes  of  multiplication. 

Thus,  in  multiplication,  if  a  and  b  stand  for  any  positive 
quantities  (see  Art.  52),     , 


^aX  +h=+ab] 
^aX—h  =  —ah 

—  aX+b  —  —ab 

—  aX~b=+ab 


Hence,  by  def- 
inition of  di-  i 
vision, 


+  a6^+6-+a...(l) 
—  ab-^—b=+a...(2) 
-ab^  +6  =  — a...  (3) 
+  a6-T--6=-a...(4) 


From  (1)  and  (2)  we  see  that  the  division  of  like  signs  gives  +. 
From  (3)  and  (4)  we  see  that  the  division  of  unlike  signs  gives  — . 
Hence,  the  law  of  signs  is  the  same  in  division  as  in  multi- 
plication. 

65.  Division  of  Monomials  in  General.  Combining  the 
results  obtained  in  Arts.  62,  63,  64,  we  have  the  follow- 
ing general  process  for  the  division  of  one,  monomial  by 
another : 

Divide  'the  coefficient  of  the  dividend  by  the  coefficient  of  the 
divisor  ; 

Obtain  the  exponent  of  each  literal  factor  in  the  quotient  by  sub- 

4: 


50  ALGEBRA. 

trading  the  exponent  of  each  letter  in  the  divisor  from  the  exponent 
of  the  same  letter  in  the  dividend ; 

Determine  the  sign  of  the  result  by  the  rule  that  like  signs  give 
plies,  and  unlike  signs  give  minus. 

Ex.  1.   Divide  27a'b'x'  by  -  9aVjx\ 

— — —  =  —  oab  ,  (Quotient, 

—  da^bx^ 

since  the  factor  a^  in  the  divisor  cancels  r*  in  the  dividend. 
Ex.  2.   Divide  a'^  " »  by  a"*  -  \ 

— — -  =a"'~'^,  Quotient. 


DIVISION  OP  A  POLYNOMIAL  BY  A  MONOMIAL. 

66.  Since  =  -  -h  -  (Distributive  Law,  Art.  33),  the 

c  c      'c 

process  of  dividing  a  polynomial  by  a  monomial  may  be 

stated  as  follows : 

Divide  each  term  of  the  dividend  by  each  term  of  the  divisor,  and 
connect  the  results  by  the  proper  signs. 

Ex.  1.   Divide  12a'x  -  lOa'y  +  6aV  by  2a^ 

12a'x-10a'y-{'6a*z''       12a'x       lOa'y       6aV       ^ 

— ^ = H —  bax  —  by 

2a'  2a'  2a'  2d'  ^ 

-\-  Za'z',  Quotient. 

Or,  we  may  arrange  the  work  more  conveniently  thus: 

2a:'J12a'x  -  lOd'y  +  6aV 

Qax  —      5y  -{-  Sd'z',  Quotient. 

Ex.  2.   Divide  6a'"  +  '  -  4a='"  +  '  -  2a^  - '  by  2a'*  -  \ 

2a"  -  V6a'"  +  '  -  4a^"  ^'  -2a'"-' 

3a'"  +  *  -  2a"  +  '-  a*"  - ',  Quotient. 


DIVISION.  61^ 

EXERCISE  11. 

Divide — 

1.  15a  by  —5a.  10.  —  m'nby  —m\ 

2.  -  3a;«  by  x.  11.  Gx'"  by  -  3x'". 

3.  Sa'a;'  by  -  Aax\  12.  7a;"2/"  "■ '  by  -  x'^/". 

4.  —  30^y  by  —  6xY  13.  —  ISx'"  "  Y""  ^Y  ^x"  "  V". 
6.  —  7a;z'  by  7zl  14.  r»  —  3x'  by  —  x. 

6.  21a:2/'^2  by  -  3a:z.  15.  20^^^  -  ^xy  by  4a;. 

7.  186c'c^'  by  -  9c'd.  1().  4a6'  -  6a'6c  by  -  2a&. 

8.  —  33:cy2'  by  l\xyh\  17.  ar»  -  x'  +  x  by  ar. 

9.  28x'2/'z'  by  —  14a;2/V.  18.  -  3a;'  +  7a;'  —  a;  by  —  ». 

19.  15x^2/  —  lOa^y  —  bxy^  by  5a;?/. 

20.  —  m  —  m'  -|-  m'  —  m*  by  —  m. 

21.  14aryz  —  2\xyh^  +  a;2/z  by  —  xyz. 

22.  9a;='"  —  6a;'"  +  llx""  by  -  3a;". 

23.  -4a;'"  +  ^  +  10a;'"  +  '-6a;"  +  'by-2a;'". 

24.  X"  +  '  -  2a;"  +  '  +  3a;"  +  '  +  a;"  by  a;"-\ 

25.  83;*"  +  '  — 16a;'"  +  '  — 4a;"*-12a;"*-^  by  -4a;'"-l 

26.  9a;'"  - '  -  6a;'"  "  ^  +  12a;'"  -  3x'"  +  ^  by  3a;"  -  \  n 


DIVISION    OP    A    POLYNOMIAL    BY    A    POLY- 
NOMIAL. 

67.  General  Method.  The  work  of  dividing  one  polyno- 
mial by  another  is  performed  to  the  best  advantage  if  we  first 
arrange  the  polynomials  according  to  the  ascending  or  de- 
scending powers  of  some  one  letter,  and  then,  in  effect,  sep- 
arate the  dividend  into  partial  dividends  (by  the  Distributive 
Law,  Art.  33),  which  ^re  then  divided  in  succession  by  the 
divisor.  7*^      <•»'*- 

Thus,  in  order  to  divide  6a;*  +  7a;'  -  3a;'  +  11a;  -  6  by  2z' 


52  ALGEBRA. 

i-Sx  —  2,  if  we  divide  the  first  term  of  the  dividend,  6a;*,  by 
the  first  term  of  the  divisor,  2^^,  and  multiply  the  quotient 
obtained,  Sx"^,  by  the  entire  divisor,  we  obtain  the  first  partial 
dividend.  If  we  subtract  this  from  the  entire  dividend  and 
proceed  in  like  manner  with  the  remainder,  we  have  a  process 
like  the  following : 

2x^  -\-  Sx  —  2,  Divisor. 


6x'  +  7x'  -  Sx'  +  llx  - 
6x'  +  dx'-6x' 

-6 

-2x'  +  Sx'-hllx- 
-2x'-Zx'+    2x 

-6 

Qx'-\-    dx- 
6x'+    dx- 

-6 

-6 

Sx^  —  x-\-  3,  Quotient, 


The  partial  dividends  into  which  the  entire  dividend  is  sep- 
arated are. 


6x*  +  9a:'  - 

-Qx' 

2x'- 

-  Sx'  +  2x 

6x'- 

h9x  - 

-6. 

These  are  divided  in  succession  by  the  divisor,  and  give  the 
partial  quotients,  Sx^,  —  x,  +3,  which  combined  form  the 
polynomial  quotient,  Sx"^  —  x-\-  3. 

Hence,  the  process  of  dividing  one  polynomial  by  another 
may  be  formally  stated  as  follows : 

Arrange  the  terms  of  both  divisor  and  dividend  according  to  the 
ascending  or  descending  powers  of  some  one  letter  ; 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  set  down  the  result  as  the  first  term  of  the  quo- 
tient ; 

Multiply  the  entire  divisor  by  the  first  term  of  the  quotient,  and 
subtract  the  result  from  the  dividend; 

Continue  the  division,  regarding  each  remainder  as  a  divi- 
dend, till  the  remainder  is  zero,  or  a  quantity  which  cannot  be 
divided. 


DIVISION.  53 

Ex.1.    Divide  4a:  +  4a;^  -  a:' by  2a:  +  2a:^  —  8x'. 
Arrange  the  terms  according  to  the  descending  powers  of  a:. 


4x^ 

-    ar»            -\-  4x\ 

2ar*  -  3a:2  +  2x,  Divisor. 

4x^- 

-  6a:*  +  43^ 
6a:*  -  52:=' 
6a:*  -  9a:»  +  6x^ 

4ar*  -  6x^  +  4x 
4x^  -  ex'  +  4x 

2a;2  +  3a:   +2,  Quotient. 

Ex.  2.   Divide  Slab'  -  206*  -  lOa'b'  +  6a*  -  a'b  by  3a'  -  5b' 
+  4ab. 

6a*  -    a^S  -  lOaW  +  31a6=*  -  206*  |  3a^  +  4a6  -  bb\  Divisor. 
6a*  +  8a=^6  -  10a^6^ 2a2  -  3a6  +  46^,  Quotient. 

-  9a^b  +  31a6=* 

-  9a^6  -  12a^6^  +  15a6^ 

+  12aW  +  16a63  -  206* 
+  12a^6^  +  16a6^  -  206* 

Ex.  3.   Divide  x^  +  y'  +  z^  +  Sx'y  +  Sxy'  hyx  +  y  +  z. 
Arranging  terms  according  to  the  descending  powers  of  x, 

x^  +  3a:V  +  3a:^2  +      y'^  ^-  ^  \x  +  y  +  z 

z^  —  yis 


7?  + '  x^y  + 

x'^z                         x'  +  2a:^  -  xz  ^ 

+  2x?y  - 
+  2a;'^.v 

x'^z  +  2>xy'^  +  ^3  +  sr*   . 
+  Ixy"^          +  "Ixyz 

- 

xH  +    xy"^  -  2xyz  -V  y^  +  2^ 
x^z  —    xz"^  —  xyz 

xy"^  +  xz'  -  xyz  +  y^  +  !? 
xy-"                      +  Tf  +  y'^z 

+  xz'  —  xyz          -  y'^z  •\-  s? 
+  xz'                     +  yz'  +  ^ 

-  xyz  -  y'^z  -  yz" 

-  xyz  -  y^z  -  yz' 

Ex.  4.  Divide  a*"  + '  -  4a'"  + '  —  27a'"  + '  +  42tt'"  by  oT  +  Sa" 

-  6a"^-^ 

^m  +  3  _  4^m  +  2  _  ^-j oT  +  ^  +  42a"'  I  g"*  +  Sa*"-^  -  6a**-' 
g"*  +  ^  +  3a*»  +  =^  -    Bg*"  +  ^ a^  -  Ta^* 

-  Ta"*  +  2  _  2ia'»  +  i  +  420"* 

-  la"^  +  2  -  21a'»  +  1  +  420** 


64  ALGEBRA. 


Divide- 


EXERCISE  12. 

1.  3a;^+7a:  +  2by  a;  +  2.      \ 

2.  Qx'  +  7a;  +  2  by  3x  +  2. 

3.  Ux"  +  xy  -  20y'  by  Sx  +  4y, 

4.  6x'  -  a;2/  -  122/'  by  2:c  -  ^y. 
6.  3a;'^  +  a;-14by  a:-2. 

6.  6:c''  -  Zlxy  +  352/'  by  2x  -  7y. 

7.  12a''  -  llac  -  36c'  by  4a  -  9c. 

8.  -  15x'  +  59a;  -  56  by  3a;  -  7. 

9.  44a;' -xy  —  By'  by  11a; -  Sy, 

^.  a'  -  W  by  a  -  2Z>./  13.  9a;'  -  49  by  3a;  +  7. 

11.  x'^  —  fhyx  —  y.  14.  125  —  64ar*  by  5  -  4x. 

12.  27ar'  +  8  by  3a;  +  2.  15.  8aV  +  y^  by  2aa;  +  ^|| 

%.  2a;'  -  9a;'  +  11a;  -  3  by  2a;  -  3. 

17.  353;^  +  47a;'  +  13a;  +  1  by  5a;  +  1. 

18.  6a'  -  17a'a;  +  14aa;^  -  3a;'  by  2a  -  3a;. 

19.  42/*  -  182/'  +  222/'  -  72/  +  5  by  22/  -  5. 

20.  c^  +  c*a;  +  c'a;'  +  c'x^  4-  ex''  +  a;^  by  c  +  a;. 
21.*  11a;  -  8a;'  +  5a;'  -  20  +  2a;*  by  a;  +  4. 

22.  4a;  +  6a;^  +  3a;'-lla;'-4by  3a;'-4. 

23.  -x?y-  11x2/'  -  2a;'2/'  +  Qx''  -  6y'  by  2a;  -  By, 

24.  42/'  +  6a;^  -  ISx'y  by  Bx'  -  2y. 

25.  X*  —  I62/*  by  a;  —  2y.  27.  x^  —  y^  by  x-^y. 

26.  x^  +  Z2f  by  x  +  2y.  28.  256a.-«  -  2/'  by  4a;'  —  y", 

29.  9a;  —  18a;'  +  8a;*  -  13a;'  +  2  by  4a;'  +  a;  —  2. 

30.  10  -  ar«  -  27a;' +  12a;* -3a;  by  a;  +  4a;' -2. 

*  Arrange  dividend  in  descending  powers  of  x. 


DIVISION.  55 

31.  22x'  -  13r'  4-  lOx'  -  18x*  +  5a;  -  6  by  x  +  Sx'  -  2. 

82.  14xy  —  16a:'2/='  +  Gx'  +  2/'  +  5^*?/  -  Qxy''  by  3x^  +  ^  -  2x2/. 

33.  Mh  -  3a'6^  -  a'h^  +  3a^  -  46^  by  a'  +  3a6  +  26^ 

34.  x'  —  /  +  2^  —  X2/2  —  2x^2  +  2yz^  hy  x  —  y  —  z, 

35.  c^  +  d^  +  n^  —  Scdn  by  c  +  d  -f-  n. 

36.  /  -  2/  +  1  by  2/'  —  22/  +  1. 

37.  2x'  +  1  —  3x*  by  1  +  2x  f  x\ 

38.  6xy  -  62/V  -  6xV  -  13x^2^  -  5xy'z  by  Zxy  +  2yz  +  3x2. 

39.  x^  -  39x  +  15  -  2x'  by  3x'  +  6x  +  x'  +  15. 

40.  4x'-9x*  +  25-14x^-x'by  2x^-x-5  +  3a^. 

41.  6x'"  +  '  -  13x"^"  +  6x'»  -  ^  by  3x"  +  ^  -  2x^ 

42.  12x*"  +  13x'"  —  X"  by  3x"  +  1. 

43.  4x**  + '  +  5x"  + '  —  X"  +  '  -  X"  +  x**  -  ^  by  x'  +  2x  +  1. 

44.  6x"  +  '  —  5x"  —  6x"-'  +  13x"-'  —  Gx'*-'  by  2x'' - 3x  +  2. 

68.  Division  of  Polynomials  having  Fractional  Coeffi- 
cients. Polynomials  having  fractional  coefficients  are  divided 
by  the  same  methods  that  are  used  for  those  having  integral 
coefficients. 

Ex.  1.    Divide  Ja'  -  ^a'b  +  ^ab''  - 16'  by  |a'  -  ^ab  +  ib\ 


ia'  -  la^b  +  ^ab'  -  ^b'  \ 

W  -  \ab  +  \b\  Divisor. 

ia'-^a'b+    lab^ 

\a  -  lb,  Qmtient. 

-ia26+    iab'-\b^ 

-Wb-\-    ^ab'-^b^ 

Ex.  2.   Divide  0.2x*  -  0.01x^2/  -  0.44xy  +  xy'  -  1.922/*   by 
0.5x'  -  0.4x2/  4-  1.22/1 

0.2x*  -  O.Olx^y  -  0.44xV  +  xy^  -  1-92^*  |  0.5x^  -  0.4x.y  +  1.2v' 
0.2x*  -  0.1 6x^?/  +  0.48xV  0.4x2  +  O.dxy  -  1.6y^ 

+  0.15x^2/  -  0.92xV  +  xy^ 
•f  O.lSx^.v  -  0.12xV  +  OMxi/^ 

-    0.8xV  +  OMxy^  -  1.92?/* 
-^jxV_+  0.64x.v^  ~  1.92iy* 


66  ALGEBRA. 

EXERCISE  13. 

Multiply — 

rg.HxV+fx-i  by|x-2. 
(^2H^'-2x  +  |byfx  +  |. 
J3i  |x»-|x^  +  |x-f  byfa^  +  f. 
,4,;  1.2x^  +  1.5x  +  6.4  by  2Ax  -  3. 
(  5.  3.6x3  _  2.8x'^  +  7.2x  -  0.32  by  1.5x  +  0.25. 
,  6.  4.5x2  -  2.8x2/  +  5.62/'  by  1.5x2  +  1.2xy  +  2.42/^. ,  , 

Divide — 

9.  Jx*  +  iix'2/  +  H^y  +  H-y'  by  |:«^  -  \xy  + 12/^ 
Q0>i  4.5x-^  -  7.1x^  -  0.4x  +  0.24  by  2.5x  +  0.5. 
ril^0.25x*  -  1.8x^  +  3.24x2  -  12.25  by  0.5x2  _  ^<^^  _  35 
l5.'4.8x*  +  0.18xV  -  Wf  +  115.19x2/3-275.22/*  by  1.6x^ 
-  2.5x2/  +  12.82^ 

Perform  the  operations  indicated — 

13.  [1 -x3  + 22/(42/' +  3x)]^(l-x  + 22/). 

14.  [1  -  2x2(x*  -  x^  +  1)  -  3x*]  -  [1  -  a:(2x2  +  1)]. 

15.  (1+  x«  +  x^O  -^  [(1  -  X  +  x')  (1  -  X*  +  x«)]. 

16.  [x^  +  (4a6  -  62)x  -  (a  -  26)  {a?  +  S^')]  ^  (x  +  26  -  a). 

17.  \x'  +  (3  -  hy  +  (c  -  36  -  2)x2  +  (26  +  3c)x  -  2c\  ^ 
(x2  +  3x  -  2). 

18.  \x\y  --z)^  yXz  -  x)  +  z\x  -y)l-^  \xXy  -  2)  +  y^z  -  x) 
+  z\x-y\, 

19.  { (a^  -  3a6)x^  +  (2a'  +  4a6  +  362)x  -  (2a6  +  56') }  -^  (ax  -  6). 


CHAPTER    VI. 
SIMPLE  EQUATIONS. 

69.  An  Equation  is  the  statement  of  the  equality  of  two 
dlgebraic  expressions. 

An  equation,  therefore,  consists  of  the  sign  of  equality  and 
an  algebraic  expression  on  each  side  of  it. 

Ex.  Sx-l=^2x  +  Z. 

70.  Members  of  an  Equation.  The  algebraic  expression 
to  the  left  of  the  sign  of  equality  is  called  the  first  memher  of 
the  equation ;  the  expression  to  the  right  of  the  sign  of  equal- 
ity is  called  the  second  memher. 

Thus,  in  the  equation  3a;  —  1  =  2x  +  3, 

the  first  member  is,  3a:  —  1 ;  the  second  member  is,  2x  +  3. 

The  members  of  an  equation  are  sometimes  called  sides  of 
the  equation. 

71.  Use  of  an  Equation.  An  equation  expresses  the  re- 
lation of  at  least  one  unknown  quantity  to  certain  given  or 
known  quantities,  the  object  in  the  use  of  the  equation  being 
to  determine  the  value  of  the  unknown  quantity  in  terms  of 
the  known. 

Thus,  in  the  above  example  x  represents  the  unknown 
quantity,  and  —  1,  2,  3  are  known  quantities. 

72.  A  Numerical  Equation  is  one  in  which  all  the  known 
quantities  are  expressed  as  arithmetical  numbers. 

Ex.  3a;- 1  =  2a; +  3. 

57 


58  ALGEBRA. 

73.  A  Literal  Equation  is  one  in  which  at  least  some  of 
the  known  quantities  are  represented  by  letters. 

Ex.  ax  +  26  =  3c  —  dx. 

a,  h,  c,  and  —  d  represent  known  quantities. 

74.  Degree  of  Equation  having"  One  Unknown  Quan- 
tity. If  an  equation  contain  but  one  unknown  quantity,  the 
degree  of  the  equation  (after  the  equation  has  been  reduced 
to  its  simplest  form)  is  determined  by  the  exponent  of  the 
highest  power  of  the  unknown  quantity  in  the  equation. 

Exs.  2a;  +  1  =  5x  —  8  is  an  equation  of  the  first  degree. 
ax  =  b'^    +  ex         is  of  the  first  degree. 
4x'-5x=20  "  second" 

'Sx'-x^   =6x  +  S      "  third     " 

An  equation  of  the  first  degree  is  also  called  a  Simple 
Equation. 

75.  The  Root  of  an  equation  is  the  number  which,  substi- 
tuted for  the  unknown  quantity  in  the  equation,  satisfies  the 
equation ;  that  is,  reduces  the  two  members  of  the  equation 
to  identical  numbers. 

Ex.  If  in  the  equation,  3x  —  1  =^  2a;  +  3, 
we  substitute  4  in  the  place  of  x, 
we  obtain  12  —   1  =   8  +  3 

11  =  11, 

and  the  equation  is  satisfied.      Hence,  4  is  the  root  of  the 
given  equation. 

76.  The  Solution  of  a  simple  equation  is  the  process  of 
finding  the  value  of  its  root. 

In  a  simple  equation  the  relation  between  the  unknown 
quantity  and  certain  known  quantities  is  given,  but  in  a 
more  or  less  complex  form,  from  which  the  value  of  the 


SIMPLE  EQUATIONS.  59 

unknown  quantity  in  terms  of  the  known  is  difficult  to  per- 
ceive. But  since  the  algebraic  symbols  which  represent  these 
quantities  may  be  rearranged  and  combined  by  the  methods 
of  Chapter  II.,  as  well  as  by  the  axioms  of  Art.  77,  the  com- 
plex relation  first  given  can  be  reduced  to  a  simple  one,  whence 
the  value  of  x  can  be  at  once  perceived. 

The  processes  of  reducing  an  equation  to  its  simplest  form 
have  been  systematized,  and  take  certain  standard  forms. 

77.  Axioms.  Besides  the  principles  given  in  Chapter  II., 
which  govern  the  use  of  algebraic  symbols  for  quantity  and 
enable  us  to  use  these  symbols  to  the  best  advantage,  there 
are  other  principles  which  are  true  of  quantity  in  general, 
and  therefore  of  algebraic  quantity.  These  principles,  -like 
those  of  Chapter  II.,  are  of  great  value  in  enabling  us  to  use 
algebraic  expressions  to  the  best  advantage. 

They  are  the  so-called  axioms : 

1.  Things  equal  to  the  same  thing,  or  to  equals,  are  equal  to  each 
other. 

2.  If  equals  he  added  to  equals,  the  sums  are  equal. 

3.  If  equals  be  subtracted  from  equals,  the  remainders  are  equaL 

4.  If  equals  be  multiplied  by  equals,  the  products  are  equal. 

5.  If  equals  be  divided  by  equals,  the  quotients  are  equal. 

6.  Like  powers  or  like  roots  of  equals  are  equal. 

7.  The  whole  is  equal  to  the  sum  of  its  parts. 

78.  Application  of  Axioms  to  the  Members  of  an  Equa- 
tion. Since  the  two  members  of  an  equation  are  equal  quan- 
tities, it  follows  from  the  axioms  of  Art.  77  that — 

The  members  of  an  equation  may  be  increased,  diminished,  mul- 
tiplied, or  divided  by  the  same  quantity,  and  the  results  will  be 


Ex.  1.   If  8a;  =  24, 
dividing  both  members  by  8  (Ax.  5), 

x  =  3. 


60  ALGEBRA. 

Ex.  2.   If  x^  +  3cc  =  x'  H- 12, 
subtracting  x^  from  each  member  (Ax.  3), 

Zx  -  12. 
Hence  cc  ^  4  (Ax.  5). 

79.  Transposition  of  Terms.     If  we  take  the  equation 

X  -}-b  =  a, 
and  we  subtract  b  from  each  member  (Ax.  3), 
x-\-b  —  b  =  a  —  b, 
or  x  =  a  —  b. 

This  result  might  have  been  obtained  at  once  from  cc  +  a 
=  6  in  a  mechanical  way,  by  transferring  -f-  b  from  the  left- 
hand  member  to  the  right-hand  member,  at  the  same  time 
changing  its  sign  to  minus. 
So,  if  we  take  the  equation, 

x  —  b=a, 
and  add  b  to  each  member,  we  obtain 
x  —  b  +  b=--a  +  bj 
or  x  =  a  +  b. 

This  result  also  might  have  been  obtained  at  once  by  trans- 
ferring —  6  to  the  opposite  member,  at  the  same  time  changing 
its  sign. 

This  process  occurs  so  often  in  simplifying  an  equation 
that  we  abbreviate  into  the  mechanical  form  and  call  it 
Transposition. 

Any  term  of  an  equation  may  be  transposed  from  one  side 
of  an  equation  to  the  other,  provided  the  sign  of  the  term  be 
changed. 

The  main  object  of  transposition  of  terms  is  to  get  all  the  terms 
containing  the  unknown  quantity  on  the  left-hand  side  of  the  equor 
tion,  and  all  the  known  terms  on  the  right-hand 


SIMPLE  EQUATIONS.  61 

I'fiis  is  the  most  important  single  step  in  simplifying  an 
equation.  This  importance  is  indicated  by  the  fact  that  the 
word  algebra  means  transposition  {al  gebre,  Arabic  words 
meaning  "  the  transposition  "). 

80.  Clearing"  of  Parentheses.  If  an  equation  contain 
quantities  in  parentheses,  it  is  necessary  first  of  all  to  remove 
the  parentheses  by  performing  the  operations  indicated  by 
them. 

81.  Changing  Signs  of  all  Terms.  The  signs  of  all  the 
terms  of  an  equation  may  be  changed.  This  may  be  regarded 
either  as  the  result  of  multiplying  both  members  of  the  equa- 
tion by  —  1,  or  as  the  result  of  transposing  all  the  terms  of 
the  equation. 

Ex.  For  —ax  =  —  b-\-c, 
write  ax  =  b  —  c. 

82.  General  Process  of  Solving  a  Simple  Equation.  This 
may  now  be  stated  as  follows : 

Clear  the  equation  of  parentheses  by  performing  the  operations 
indicated  by  them; 

Transpose  the  unknown  terms  to  the  left-hand,  side  of  the  equa- 
tion^ the  known  terms  to  the  right-hand  side; 

Collect  terms; 

Divide  both  members  by  the  coefficient  of  the  unknown  quantity. 

Ex.  1.   Solve  the  equation,  3x  —  7  =  14  —  Ax. 
Transposing  the  terms  —  7  and  —  Ax^  3a;  +  4a;  =  14  +  7 
Collecting  terms,  Ix  =  21 

Dividing  by  7,  ^c  =  3,  Root. 

Verification.    Substituting  3  for  a;  in  3x  —  7  =  14  —  Ax 

3X3-7  =  14-4X3 
9-7  =  14-12 
..      ,-   »  -,    -      .     .      2=2. 


62  ALGEBRA. 

Ex.  2.   Solve  x(x  --  2)  =  x(ix  +  4)  -  S(x  -  3). (1) 

Removing  parentheses,  x^  —  2x  =  x'^  +  4a;  —  3a;  +  9 
Transposing  terms,  x^  -  x^  —2x  —  4a;  +  3x  =  9 
Collecting  terms,  —  3a;  =  9  .  .  (2) 

Dividing  by  —  3,  x=  —  S,  Boot. 

Verification.     Substituting  —  3  for  a;  in  equation  (1), 
-3(-3-2)  =  -3(-3  +  4)  -3(-3-3) 
-3(-6)  =  -3(l)-3(-6) 

15--3  +  18 

15  =  15. 

The  value  of  the  processes  employed  in  the  solution  of  an 
equation — viz.  transposition,  etc. — is  realized  when  we  com- 
pare the  ease  with  which  the  value  of  x  is  perceived  from 
equation  (2),  with  the  difficulty  in  assigning  immediately  in 
equation  (1)  a  value  for  x  which  will  satisfy  that  equation. 

EXERCISE  14. 

Solve  the  following  equations : 

1.  3a;=15.  9.  — 7a;  =  0. 

2.  2a;  =  -  6.  10.  4a;  -  5  =  1  +  3a;. 

3.  -13a;  =  26.  11.  2a; -7  =  8  + 5a;. 

4.  _6a;=-12.  12.  2a;  -  3(a;- 3)  +  2  =  0. 

5.  3a;  =  -  5.  13.  7(2  -  3a;)  =  2(7  -  8a;). 

6.  — 2a;  =  ll.  14.  x' -  a;(a;  +  5)  =  a;  +  12. 

7.  -  5a;  =  _  13.  15.  3  _  2(Sx  +  2)  =  7. 

8.  4a;=  — 1.  16.  2a; -  (a;  +  5)  =  4a;. 

17.  (x  -  1)  (a;  +  3)  =  (x  -  4)  (a;  +  2). 

18.  3a;  -  4a; +  10  + 5a;  =  0. 

19.  a;(2a;  +  1)  -  2a;(a;  +  3)  =  7. 


SIMPLE  EQUATIONS,  ,  63 

20.  3(a:  -  1)  (a;  +  1)  =  x(Sx  +  4). 

21. 4(x-sy  =  {2x  +  iy. 

22.  8(2;-3)-(6-2a:)=2(a:  +  2)-5(5-a;). 

23.  5x -  (3a; -  7)  -  {4  -  2x-  (6a:  -S)]=  10. 

24.  a;  +  2-[a:-8-2S8-3(5-a;)-a;n=0. 

25.  2(a;  +  1)  (2a;  -  1)  +  2|a;  -  (a;  +  3)  (2a;  -  1)5  =  -  32. 

26.  2a;(a;  -5)-\x'  4-  (3a;  -  2)  (1  -  a;)  \  -  (2a;  -  4)1 

27.  (x  +  1)^  -  2\(x  -  1)^  ~  3(a;  +  2)'^S  -=  3(a;  +  4)  (2a;  -  4) 
^(a;^-5). 

28.  8a;''  +  13a;  -2\x'-  3[(a;  -  1)  (3  +  a;)  -  2(a;  +  2)^]S  =  3. 

SOLUTION   OF  PROBLEMS. 

83.  The  General  Method  of  stating  and  solving  will 
first  be  illustrated  by  an  example  similar  to  that  given  in 
Art.  1. 

James  and  John  together  have  $18.  If  James  has  twice 
as  many  dollars  as  John,  how  many  dollars  has  each 
boy? 

By  solution  of  the  problem  is  meant,  of  course,  finding 
the  value  of  the  unknown  quantity  or  quantities  of  the 
problem.  The  first  thing  to  be  done,  therefore,  is  to  de- 
termine what  are  the  unknown  quantities  or  numbers. 
In  the  given  problem  there  are  evidently  two  unknown 
quantities  to  be  determined — first,  the  number  of  dollars 
which  James  has;  second,  the  number  of  dollars  which 
John  has. 

The  method  of  procedure  is  then  to  state  the  relation  of 
the  unknown  quantities  to  the  known  quantities  in  the  form 
of  an  equation,  which  is  afterward  solved. 

As  a  rule,  when  there  are  two  unknown  quantities  in  a 
problem,  it  is  more  convenient  to  represent  the  smaller  of 
them  by  x. 


64 


ALGEBRA. 

Let 

X  =  number  of  dollars  John  has. 

Therefore, 

2x=       " 

James  has. 

Hence, 

x-{-2x=       " 

both  have. 

But 

18  also  =       " 

a 

Hence,  by 

use  of  Axiom  1,  Art.  77, 

.x  +  2x  =  lS, 
3:^-18, 

a:  =^  6,  number  of  dollars  John  has. 

2x-12,      " 

James  has. 

It  is  to  be  noticed  particularly  that  we  let  x  equal  a  definite 
number^  not  a  vague  quantity. 

We  do  not  let  a:  =  the  money  which  John  has, 

nor  X  =  what  John  has, 

but  let  X  =  number  of  dollars  which  John  has. 

In  solving  problems  the  student  will  find  it  necessary  to 
study  each  problem  carefully  by  itself,  as  no  rule  or  method 
can  be  found  which  will  cover  all  cases. 

The  following  general  directions  will,  however,  be  found  of 
service: 

By  study  of  the  problem  determine  what  the  unknown  quantity 
or  quantities  are  whose  values  are  to  be  obtained; 

Let  X  equal  one  of  these  expressed  as  a  number; 

State  all  the  other  unknown  quantities  which  are  either  to  be 
determined  or  to  be  utilized  in  the  process  of  the  solution,  in  terms 
of  x; 

Obtain  an  equation  by  the  use  of  a  principle,  such  as  the  whole 
is  equal  to  tJce  sum  of  its  parts,  or  things  equal  to  the  same  things 
are  equal  to  ea,ch  other ; 

Solve  the  equation,  and  find  the  value  of  each  of  the  unknown 
quantities. 


SIMPLE  EQUATIONS.  65 

Ex.  1.  James  and  John  together  have  $24,  and  James  has 
8  dollars  more  than  John ;  how  many  dollars  has  each  ? 

Let  X  =  number  of  dollars  which  John  has. 

Since  James  has  8  dollars  more  than  John, 

a;  +  8  ~  number  of  dollars  which  James  has. 
.•.x  +  (a;  +  8)=       "  "  "      both  have. 

But  24-       "  "  "         "        " 

Hence,  by  Axiom  1, 
a;  -f  a;  +  8  =  24 
2a:  =  16 
a;  =  8,  number  of  dollars  which  John  has. 
a; +  8 -=16,     "  "  "   *  James  has. 

Ex.  2.  The  sum  of  two  numbers  is  36,  and  three  times  the 
greater  number  exceeds  four  times  the  less  number  by  10. 
Find  the  numbers. 


Let  a:  =  the  less  number. 

If  the  sum  of  the  numbers  is  36  (as  the 
line  AB),  and  one  part  is  x  {AC),  then 
the  other  part  will  be  36  -  a;  {CB). 

.  * .       36  —  a:  =  the  greater  number. 


36 


I 
-B 


36 


.  * .  3(36  —  x)  =  three  times  the  greater  number. 

4a;  =  four      "        "   less  '' 

But  (three  times  the  greater  number)  —  (four  times  the  less)  =  10. 
.  • .       3(36  -  a:)  -  4a;  =  10. 
Hence,  108  -  3a;  -  4a;  =  10 

-  7a;  =  -  98 

a;  =  14,  the  less  number. 
36  —  a;  =  22,  the  greater  number. 
Let  the  student  verify  these  results. 

Problems  may  frequently  be  solved  in  more  than  one  way. 
Thus,  Ex.  1  above  may  be  stated  and  solved  as  follows : 
5 


66  ALGEBRA. 

Let  X  =  number  of  dollars  which  John  has. 

Then  24  —  a;  =       ''         ''  ''      James  has. 

Also,  (number  of  James's  dollars)  —  (number  of  John's  dollars)  =  8  ; 
that  is,  (24  -  a;)  -  a;  =  8 

24  -  a;  -  a:  =  8 

-  2a;  =  -  16 

a;  =  8,  number  of  John's  dollars. 
24  -  a;  =  16,         *'         James's     " 


EXERCISE  15. 

ORAL. 

1.  A  has  X  marbles,  and  B  has  twice  as  many  ;  how  many  has  B  ?    How 
many  have  both  ? 

2.  There  are  100  pupils  in  a  school,  of  which  x  are  boys ;  how  many  are 
girls  ? 

3.  If  I  have  x  dollars,  and  you  have  three  dollars  more  than  twice  as 
many,  how  many  have  you?     How  many  have  we  together? 

4.  Two  boys  together  solved  a  examples :  the  one  did  x  ;  how  many  did 
the  other  solve? 

5.  The  difierence  between  two  numbers  is  15,  and  the  less  is  x ;  what  is 
the  greater?     What  is  their  sum? 

6.  If  n  is  a  whole  number,  what  is  the  next  larger  number?    The  next 
less? 

7.  Write  three  consecutive  numbers,  the  least  being  x.     Write  them  if 
the  greatest  is  y. 

8.  John  has  x  dollars,  and  James  has  seven  dollars  less  than  three  times 
as  many  ;  how  many  has  James? 

9.  If  I  am  x  years  old  now,  how  old  was  I  ten  years  ago?     a  years  ago? 
How  old  will  I  be  in  c  years  ? 

10.  A  man  bought  a  horse  for  x  dollars,  and  sold  it  so  as  to  gain  a  dol- 
lars ;  what  did  he  receive  for  it? 

11.  A  man  sold  a  horse  for  $200,  and  lost  x  dollars  ;  what  did  the  horse 
cost? 

12.  If  a  yard  of  cloth  cost  m  dollars,  what  will  x  yards  cost? 

13.  If  a  boy  ride  a  miles  an  hour,  how  far  will  he  ride  in  c  hours? 

14.  A  bicyclist  rides  x  yards  in  y  seconds ;  how  far  will  he  ride  in  one 
second?    In  n  seconds? 


SIMPLE  EQUATIONS.  67 

15.  How  many  hours  will  it  require  to  walk  x  miles  at  a  miles  an  hour  ? 

16.  A  man  has  a  dollars,  and  b  quarters  ;  how  many  cents  has  he? 

17.  How  many  dimes  in  x  dollars  and  y  halves  ? 

18.  I  have  x  dollars  in  my  purse  and  y  dimes  in  my  pocket ;  if  I  give 
away  fifty  cents,  how  much  have  I  remaining? 

19.  By  how  much  does  oO  exceed  x  ? 

20.  Express  the  sum  of  the  squares  of  two  consecutive  even  numbers  if 
the  larger  is  x. 

21.  A  gentleman  is  out  x  hours,  of  which  he  rides  a  hours  at  the  rate  of 
eight  miles  an  hour,  and  walks  the  rest  of  his  time  at  the  rate  of  three 
miles  an  hour ;  how  far  did  he  ride  ?    How  far  did  he  walk  ? 


EXERCISE  16. 

1.  A  boy  has  three  times  as  many  marbles  as  his  brother, 
and  together  they  have  48;  how  many  has  each? 

2.  A  and*B  pay  $100  taxes;  if  A  pays  $22  more  than  B, 
what  does  each  pay? 

3.  John  solved  a  certain  number  of  examples,  and  William 
did  12  less  than  twice  as  many ;  both  solved  96.  How  many 
did  each  solve  ? 

4.  Three  boys  earned  together  $98;  if  the  second  earned 
$11  more  than  the  first,  and  the  third  $28  less  than  the  other 
two  together,  how  many  dollars  did  each  earn  ? 

5.  A  man  walked  15  miles,  rode  a  certain  distance  in  a 
coach,  and  then  took  a  boat  for  twice  as  far  as  he  had  pre- 
viously traveled;  altogether,  he  went  120  miles.  How  far 
did  he  ride  by  boat? 

6.  Find  three  consecutive  numbers  whose  sum  is  84. 

7.  The  sum  of  two  numbers  is  92.  and  the  larger  is  3  less 
than  four  times  the  less;  find  the  numbers. 

-8.  The  sum  of  three  numbers  is  50 :  the  first  is  twice  the 
second,  and  the  third  is  16  less  than  three  times  the  second ; 
find  the  numbers. 

9.  A  farmer  paid  $94  for  a  horse  and  cow;  what  did  each 
cost,  if  the  horse  cost  $13  more  than  twice  as  much  as  the 
cow? 


G8  ALGEBRA. 

10.  Distribute  $485  among  A,  B,  and  C  so  that  B  and  C 
each  get  twice  as  much  as  A. 

11.  Divide  the  number  35  into  two  parts,  such  that  three 
times  the  smaller  shall  be  equal  to  twice  the  larger. 

12.  Find  five  consecutive  numbers  whose  sum  shall  be  3 
less  than  six  times  the  least. 

13.  The  difference  between  two  numbers  is  6,  and  if  3  be 
added  to  the  larger  the  sum  will  be  double  the  less;  find 
the  numbers. 

14.  Three  men  rent  a  store  for  $500:  the  first  is  to  pay- 
twice  as  much  as  the  second,  and  the  second  $60  more  than 
the  third;  how  much  does  each  pay? 

15.  Divide  $4500  among  two  sons  and  a  daughter  so  that 
each  son  gets  $150  less  than  twice  the  daughter's  share. 

16.  A  father  is  four  times  as  old  as  his  son,  and  the  differ- 
ence of  their  ages  is  24  years  less  than  the  sum ;  how  old  is 
each? 

17.  A  man  is  twice  as  old  as  his  daughter,  who  is  5  years 
younger  than  her  brother,  and  the  combined  ages  of  all  three 
are  109  years ;  what  is  the  age  of  each  ? 

18.  A  father  is  now  twice  as  old  as  his  son ;  21  years  ago  he 
was  three  times  as  old.     How  old  are  they  now  ? 

19.  Find  two  numbers  whose  difference  is  14,  such  that  the 
greater  exceeds  twice  the  less  by  3. 

20.  Find  three  consecutive  odd  numbers  whose  sum  is  63. 

21.  The  greater  of  two  numbers  is  5  more  than  the  less,  and 
five  times  the  less  exceedr.  three  times  the  greater  by  3;  find 
the  numbers. 

22.  A  man  had  five  sons,  to  whom  he  gave  $56,  giving  to 
each  $5  less  than  twice  the  amount  his  next  younger  brother 
received ;  what  did  each  receive  ? 

23.  It  is  required  to  divide  75  into  two  such  parts  that  three 
times  the  greater  exceeds  seven  times  the  less  by  15. 

24.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  43;  find  the  numbers. 


SIMPLE  EQUATIONS.  69 

25.  The  difference  of  the  squares  of  two  consecutive  even 
numbers  is  60;  find  the  numbers. 

26.  The  joint  ages  of  father  and  son  are  64  years :  if  the 
age  of  the  son  were  doubled,  he  would  still  be  four  years 
younger  than  his  father;  find  the  age  of  each. 

27.  Two  bicyclists,  A  and  B,  start  respectively  from  New 
York  and  Philadelphia,  90  miles  apart,  and  ride  toward  each 
other ;  A  rides  8  and  B,  12  miles  per  hour.  How  long  and 
how  far  will  A  ride  before  meeting  B? 

28.  A  man  walks  to  the  top  of  a  mountain  at  the  rate 
of  2  miles  an  hour,  and  back  down  at  4  miles  an  hour; 
if  he  is  out  6  hours,  how  far  is  it  to  the  top  of  the  moun- 
tain? 

29.  How  far  into  the  country  will  a  man  go  who  rides  out 
at  the  rate  of  9  miles  an  hour,  walks  back  at  6  miles  an  hour, 
and  is  gone  10  hours  ? 

30.  A  boy  was  engaged  to  work  50  days  at  75  cents  each 
day  for  the  days  he  worked,  and  to  forfeit  25  cents  every  day 
he  was  idle.  On  settlement  he  received  $25.50 ;  how  many 
days  did  he  work? 

31.  Five  years  ago  A  was  twice  as  old  as  B,  but  10  years 
ago  he  was  three  times  as  old  as  B;  how  old  is  each 
now? 

32.  A  man  is  30  years  older  than  his  son,  and  10  years  ago 
he  was  three  times  as  old ;  what  is  the  age  of  each  ? 

33.  Twenty  yards  of  silk  and  30  yards  of  cloth  cost  $99, 
and  the  silk  cost  three  times  as  much  per  yard  as  the  cloth ; 
how  much  did  each  cost  per  yard  ? 

34.  A  merchant  paid  a  bill  of  $72  with  dollar,  two-dollar, 
and  five-dollar  bills,  paying  the  same  number  of  each ;  how 
many  of  each  did  he  use  ? 

35.  How  can  $2.25  be  paid  in  five-  and  ten-cent  pieces  so  as 
to  use  the  same  number  of  each  ? 

36.  How  can  $5.95  be  paid  in  dimes  and  quarters,  using 
the  same  number  of  each? 


70  ALGEBRA. 

87.  A  purse  contains  $10.50  in  dollar  bills  and  quarters,  but 
there  are  twice  as  many  quarters  as  bills ;  how  many  are  there 
of  each  ? 

38.  Twenty  coins,  dimes  and  half  dollars,  make  together 
$8.80;   how  many  are  there  of  each? 

39.  A  gentleman  gave  $525  to  his  son  and  daughter,  so  that 
for  every  dime  the  daughter  received  the  son  got  a  quarter ; 
how  much  did  each  receive? 

40.  A  person  was  desirous  of  giving  80  cents  apiece  to  some 
beggars,  but  found  he  had  not  money  enough  by  80  cents ;  he 
gave  them,  therefore,  20  cents  each,  and  had  30  cents  remain- 
ing.    Required  the  number  of  beggars. 

41.  A  sum  of  money  is  divided  among  8  persons,  A,  B,  and 
C,  so  that  A  and  B  have  $79,  B  and  C  have  $70,  and  A  and  C 
$75 ;  how  much  has  each  ? 

42.  A  woman  sold  12  new  baskets  for  $3.  For  a  part  she  got 
20  cents  each,  and  for  the  rest  32  cents  each ;  how  many  of 
each  grade  did  she  sell? 

43.  A  certain  flag-pole  is  69  feet  long,  and  has  12  feet  of  its 
length  in  water:  the  part  in  air  is  3  feet  more  than  five  times 
the  length  of  the  part  in  earth;  what  is  the  length  of  the 
part  above  water? 

44.  B  had  $5  more  than  three  times  as  much  as  A,  but  he 
gave  A  $9,  and  now  he  has  a  dollar  less  than  twice  A's  sum ; 
how  much  did  each  have  at  first? 

45.  There  is  a  fish  whose  tail  weighs  9  pounds ;  his  head 
weighs  as  much  as  his  tail  and  half  his  body ;  and  his  body 
weighs  as  much  as  his  head  and  his  tail.  What  is  the  weight 
of  the  whole  fish  ? 

46.  A  set  out  from  a  town,  P,  to  walk  to  Q,  45  miles 
distant,  an  hour  before  B  started  from  Q  toward  P. 
A  walked  at  the  rate  of  4  miles  an  hour,  but  rested 
2  hours  on  thfi  way;  B  walked  at  the  rate  of  8  miles 
an  hour.  How  many  miles  did  each  travel  before  they 
met? 


BEVIEW.  71 

EXERCISE  17. 

REVIEW. 
Add— 

l.^x^  +  ^xp  -  \y\  2^2  _  2,2  _  2^^^   2a;2  -  xy  -  \y\ 

2.  \^  -  x"  ^  \x,  Ix"  -  f  a:  4-  f ,  \x  -  \  -  \7?,  \7?  +  \x^. 

3.  l.Sx-'  -  Z.ixy  +  0.6^2^  3.2a;2  -  S.l?/^  +  1.5^^^  j  7^.^  -  2/'  -  0.7a:'. 

4.  1.6  +  1.1x3  _  2.2x2,  hXx"  -  0.7a;  -  2.3a:3^  1  +  0.2a;^  -  2.9a;'  +  1.6a;. 


l.la;'^  +  2a;2/'  -  1.5^/^. 


Subtract— 

h.  ^- 

|a;2  +  f  a;  - 

h\  from 

|x' 

-|a;2  + 

|a;  + 

f. 

6.  2.7a;3 

-  0.4a;2^  - 

1.3a;y  + 

y. 

from  3a;3 

-  1.: 

La;^ 

Multiply— 

7.  \x^- 

-  f  a;  +  4  by 

%x  +  2. 

8.  1.5a;  -  0.4^  by  2.4a;  +  1.5?/. 

9.  fx'  —  ax  -V  \a^  by  fx'  +  |aa;  +  \o^. 

10.  fa;'  -  fa;  +  1  by  fa;'  +  fa;  -  1. 

11.  3.2a'  -  2.3a6  +  5.26'  by  1.5a  +  2.56. 

12.  0.4a;'  -  \.%xy  -  2.8^/'  by  0.5a;  -  \.hy. 

13.  4.5a;3^  -  1.2a;'?/'  -  hAxy^  by  0.4a;'^  -  ^.hxy^, 

14.  3.2a;'  -  4.5a;^  +  1.8^'  by  1.5a;  -  3.5^. 

Divide — 

15.  6a;3  _  4^3  _  2a;2  -  I6a;'^  +  14a;^'  +  ^xy  -  2^'  by  3a;  -  2^  -  1. 

16.  x'^  -  5a;V  +  13a;*^  -  2a;V  -  l^a;'?/^  +  9?/'  by  a;'  +  2a;y  -  Zy\ 

17.  2  —  a;  by  1  +  a;  to  five  terms  in  the  quotient. 

18.  1  —  a;  +  a;'  by  1  +  a;  +  a;'  to  five  terms. 

19.  a;^  —  15  by  a;'  +  a;  —  1  to  five  terms. 

20.  1  by  1  —  2a;  —  3a;'  to  five  terms. 

21.  -V-a^  -  \x^y  +  2a;?/'  -  -i^y^  by  \x  -  \y. 

22.  -i^x'^  -  Ix^y  +  if^V  +  \^y'  by  \x  +  \y. 

23.  36a;'  +  i^'  +  i  -  4a;?y  -  6a;  +  \y  by  6a;  -  I3/  -  J. 

24.  2.4a;3  _  o.l2a;'^  +  4.32^/=*  by  1.5a;  +  1.8y. 


72  ALGEBRA. 

25.  8.4a:*  -  l.ea:^  -  lO.Sa:^  +  10.2a;  -  3.9  by  2Ax''  +  1.6a;  -  2.6. 

26.  5a:*  -  6.S5x^^  +  SMxy^  -  OM^/*  by  2.5a:  -  0.3^. 

Simplify — 

27.  6z  -h  [43  -  {8a:  -  {2z  +  4a:)  -  22a:}  -  7a:]  -  [7a:  +  {140  -  {4z 
-  5a:)}]. 

28.  a2(6  -  c)  -  62^a  -  c)  +  c2(a  -  6)  -  (a  -  6)  (a  -  c)  (6  -  c). 

29.  2a:  -  [  -  3(a:  -  ^)  +  {(a:  -  2?/)  -  2(.a:  4  3^/)  -a:}  -  2(^  -  2a:)]. 

30.  6{a  -  2[6  -  3(c  +  d)]}  -  4{a  -  3[5  -  4(c  -  d)]}. 

31.  l-2{-[-  (a: -2/)]  -a:}  +  2{  -  2[- 1  -  (a: -?/)]- 1}. 

32.  Sa'ia  -  b)  -  (36  -  2a)  [a(26  -  3a)  -  {a  +  by]  +  6(6  +  a)^. 

Solve  and  verify — 

33.  (2a:  -f-  1)  (a;  -  3)  +  7  =  a:  -  2(a:  -  4)  (2  -  x). 

34.  7a:  -  2(a;  -  1)  (2  -  a:)  -  17  =  a:(3a:  +  7)  -  (a:  +  1)^. 

35.  2a:2  -  3a:  -  2(2a:  +  1)^  +  (2a;  -  3)  (3a:  +  2)  =  8. 

36.  3a:2  „  ^5^  _  [4  _  (^^  _  i)  ^2x  -  3)  -  7a:]  +  {x  -  3)^  =  0. 

37.  5a:  +  1  -  2{2a:  -  3[a:  -  (a:  +  1)  (a:  +  3)]  -  3(a:  +  2f}  =  0. 

38.  What  is  the  dividend  when  the  quotient  is  x^  +  2x^  +  7a:  +  20,  the 
remainder  62a:  +  59,  and  the  divisor  a:^  —  2a:  —  3  ? 

39.  What  is  the  divisor  if  the  quotient  is  x^  +  3x,  the  dividend  a:^  —  8, 
and  the  remainder  9a:  —  8  ? 

40.  If  a:  .=  -  f  and  2/  =  -  f ,  find  the  value  of 

(3a:  -  2yy  {9x''  +  4^')  -  Q[y  -  x)  l/6a:^(a:  +  2y'  +  ^). 


CHAPTER   VII. 

CASES  OF  ABBREVIATED   MULTIPLICATION  AND 
DIVISION. 

ABBREVIATED   MULTIPLICATION. 

84.  Value  of  Abbreviated  Multiplication.  In  certain 
cases  of  multiplication,  by  observing  the  character  of  the 
expressions  to  be  multiplied,  it  is  possible  to  write  out  the 
product  at  once,  without  the  labor  of  the  actual  multiplica- 
tion. Almost  all  the  multiplication  of  binomials,  and  that 
of  many  trinomials,  will  fall  under  these  cases^  and  by  the 
use  of  the  abbreviated  methods  at  least  three-fourths  of  the 
labor  of  multiplication  in  them  will  be  saved.  The  student 
should  therefore  master  them  as  thoroughly  as  he  has  done 
the  multiplication  table  in  arithmetic. 

85.  I.  Square  of  the  Sum  of  Two  Quantities. 
Let  a  +  6  be  the  sum  of  any  two  algebraic  quantities. 
By  actual  multiplication,  a-{-b 


u^  +  ah 

-{-ah    +6' 
a^  +  2ah  +  6^*,  Product^ 

Or,  in  brief,  (a  +  by  =  a'-{-  2ab  +  6^ 

which,  stated  in  general  language,  is  the  rule— 

The  square  of  the  sum  of  two  quantities  equals  the  square  of  the 
first,  plus  twice  the  product  of  the  first  by  the  second,  plus  the  square 
of  the  second. 


74  ALGEBRA. 

Ex.  (2a;  +  Syf  =  4x'  +  12xy  +  dy\  Product. 
Since  the  square  of  2x  is  4x'^, 

twice  the  product  of  2x  and  Sy  is  12a;y, 
and  the  square  of  Sy  is  %^ 

86.     II.  Square  of  the  Difference  of  Two  Quantities. 

By  actual  multiplication,  a  —  b 

a  —  b 


d  —  ab 

-ab    +y 
a'  -  2ab  +  h\  Product 

Or,  in  brief,  (a  -  by  =-a^-  2ab  +  b\ 

wliich,  stated  in  general  language,  is  the  rule — 

(The  square  of  the  difference  of  two  quantities  equals  the  square  of 
e  first,  minus  twice  the  product  of  the  first  by  the  second,  plus  the 
square  of  the  second.^^ 

Ex.  (2x-Zy)'^^x^-l2xy^-'dy\  Product. 

87.     III.  Product  of  the   Sum  and  Difference  of  Two 
Quantities. 

By  actual  multiplication,    a  +  b 

a  —  b  ' 


a^  +  ab 
-ab-b' 

a^  —  b'\  Product. 

Or,  in  brief,  (a  -^b)  (a —  b)  =  a^  —  b"^, 

which,  stated  in  general  language,  is  the  rule — 

The  product  of  the  sum  and  difference  of  two  quantiiies  is  the 
difference  of  their  squares. 

Ex.  (2x  +  Sy)  (2a;  -  Sy)  =  Ax'  -  %',  Product. 


ABBREVIATED  MULTIPLICATION.  76 

EXERCISE  18. 

Write  by  inspection  the  values  of — 

1.  (n  +  2/)'.  11.  (x^z)(x~z\ 

2.  (c-xy.  12.  (2/ -3)  (2/ +  3). 
S.(2x~yy.  13.  (3x-2/)(3x  +  2/). 

4.  (Sx-2yy.  14.  (7x  +  4y)(7x-4y-). 

6.  (5a; +  1)'.  15.  (x' -2)  (x' +  2), 

6.  (x'  +  iy.  16.  {aT^  -  h'y)  (ax' +  b'y), 

7.  (x  -  2/'')'.  17.  (1  -  lla^O  (1  +  liar'). 

8.  (l  —  7yy.  18.  (2x"  +  52/™)  (2x'*  -  62/"*). 
^  (3a:*  +  5r')^  19.  (5a;"  -  32/"z"*)l 

10.  (6xV-ll2/'zT  20j:4a;yz'"  +  92/"*)'^ 

88.  Special  Case  under  III.  In  applying  any  of  the 
above  abbreviated  methods  of  multiplication  we  may  have 
parentheses  containing  two'  or  more  terms  used  as  a  single 
quantity.  This  is  of  especial  ^importance  in  obtaining  the 
product  of  a  sum  and  difference. 

Ex.  1.   Multiply  x  +  (ia'h  b)  by  a;  -  (a  +  6). 
We  have 
[a;  +  (a  +  6)]  [x  -  (a  +  6)]  =  ar^  -  (a  +  by,  by  III. 

=  a;'-(a'  +  2a6  +  60,by  I. 
=  a;'  -  a'  -  2a6  —  6^  Product 

It  is  frequently  necessary  to  re-group  the  terms  of  trino- 
mials in  order  that  the  multiplication  may  be  performed  by 
the  above  method. 

Ex.  2.   Multiply  x  +  y  —  zhyx  —  y-\-z. 

(x  +  y  -  z)  (x  -y  +  z)  =--[x  +  (y  -  z)'][x  -  (y  -  z)] 
=  x^-(2/-z)^byIIL 
=  x'.-(y'-2yz  +  z'),hylL 
=  x^  —  2/*  +  22/z  —  z^j  Produd, 


76  ALGEBRA, 

EXERCISE  19. 

Write  by  inspection  the  product  of — 

1.  [(a  +  6)  +  3][(a  +  />)-3]. 

2.  [2x-l-y-][2x-l  +  yl 

8.  [4-(a:+l)][4  +  (^+l)]. 

4.  [a +(6 -2)]  [a -(6 -2)]. 

5.  '(2x  +  Sy  +  1)  (2x  -Sy-  1). 
■  V      1^,  (x^  +  Sx-2)  (x'  +  3x  +  2). 

^.(4    --x-y)(4  +  xi-y). 

h.  (Sx'  -  2:c  +  1)  (Sx'  +  2x  -  1). 

9.  (x^  -  xy  +  7/^)  (x'^  +xy  +  y^). 

10.  (a'  +  a  +  l)(a'-a  +  l). 

11.  (2x'-Sx-5)(2x'  +  Zx-5). 

12.  (2a;'^  +  5xy  -  f)  (2x^  -  bxy  -  y^). 

13.  (aa:'  -bx-{-  2c)  (ax'^  +  6a;  -  2c). 

14.  (x'  ~\-xy  —  2/')  (a;2  -xy  —  2/'). 

15.  (6ft^  -  3a  -  2)  (6a'  +  3a  -  2). 

16.  [(a  +  6)  -  (c  -  1)]  [(a  +  6)  +  (c  - 1)]. 

17.  [(a:'  -f  2/^)  +  (xy  +  1)]  [(^.^  +  y')  -  (xy  +  1)}     ' 

18.  (x-y  +  z-l)(x  +  y^z  +  l). 

19.  (x'-2a;'-a;-2)(a;^H-2x'  +  a;-2). 

/ 

Simplify — 

20.  (3a  -  1)'  +  (2  -  3a)  (2  +  3a). 

21.  (2x  -  7y)  (2x  +  7y)  -  4(x  -  2yy  +  132/(52/  -  x), 

22.  (3a;'  +  5)'  +  a;'(10  -  3a;)  (10  +  3a;)  -  (5  +  13a;7. 

23.  (a-c  +  1)  (a  +  c-l)-(a-l)'  +  2(c-l)l 

24.  (x  +  y  —  xy)  (x  —  y  —  xy)  +  x'y  -(x-  y')  (x  +  2/'). 

25.  (x'  -  x-^)  (x'  +  x-^)  i-  (^-t  xr\-  x')  (x-x"  i-S), 


ABBREVIATED  MULTIPLICATION.  77 

89.     IV.  Square  of  any  Polynomial. 

By  actual  multiplication, 

a  +  b      +  c 
a  +  b     -\-  c 


+  ab  +b'-\-bc 

+  ac  -i-bc    +c^ 

a'-h  2ab  +  2ac  +  6'  +  2bc  +  c\ 

or,  in  brief,  (a  +  6  +  c)^  =  a'*  +  6'  +  c'  +  2ab  +  2ac  +  26c. 
In  like  manner  we  obtain 

ia-hb-{-c  +  dy  =  a'  +  b'  +  c'  +  (r  +  2ab  +  2ac  +  2ad  +  2bc 
+  2bd  +  2cd. 

It  is  seen  that  each  of  these  results  consists  (1)  of  the 
square  of  each  term  of  the  polynomial,  and  (2)  of  other 
terms  formed  by  taking  twice  the  product  of  each  term  into 
each  term  which  follows  it.  It  is  also  perceived  that  this 
method  of  forming  the  square  of  a  polynomial  will  hold 
good  no  matter  how  many  terms  there  are  in  the  polyno- 
mial. For  if  we  examine  the  above  process  of  the  multipli- 
cation of  a  +  6  +  c  by  itself,  we  see  that  not  only  is  each  term 
multiplied  by  itself,  but  also  a  of  the  multiplicand  is  multi- 
plied by  b  of  the  multiplier ;  and,  vice  versa,  b  of  the  multi- 
plicand is  multiplied  by  a  of  the  multiplier,  and  so  for  any 
other  pair  of  terms,  and  that  this  list  exhausts  the  partial 
products.     Hence,  in  general, 

The  square  of  any  polynomial  equals  the  sum  of  the  squares 
of  all  the  terms,  together  with  twice  the  product  of  each  term 
into  each  term  which  follows  it. 

Ex.  {a  —  2b-{-c—3xy=^a^-\-ih^+c^-\-9x^—^ab-]-2ac 
—  6ax  —  46c  +  126a;  —  6cx. 


78,  ALGEBRA. 

EXERCISE   20. 

Write  by  inspection  the  values  of — 

1.  (2a;  +  2/  +  1)'.  8.  (2a}  +  5a  -  3)^ 

2.  {x-2y  +  2z)\  9.  (x-y  +  z-l)\ 

3.  (3x  -  22/  -  5)^  10.  (2a;  +  3^/  -  4z  -  5)^ 
JL  (2a-6  +  3c)\  ll  (3a;» - 4x^  +  a: - 2)^ 
^^a;-22/-3zy.  12.  {2a-h')(2a  +  h'), 

6.  (4rr  +  32/-l)^  13.  (2a'^>-56c7. 

7.  {x'-x  +  l)\  14.  (a-6  +  7)(a  +  6-7) 

15.  (2a:'  -  7a:  +  11)  (2a:'  +  7a:  +  11). 

16.  (5a:'-6x-3)l 

17.  (a:' +  3a: -5)  (a:' -3a: -5). 

18.  (3a'  -  5a&  +  h')  (3a'  -  5a6  -  6'). 

19.  (7x'  -  5a:  +  4)  (7x'  +  5a:  -  4). 

20.  [(a'  +  26')  +  (2a6  -  1)]  [(a'  +  26')  -  (2a6  - 1)]. 

90.    V.  Product  of  Two  Binomials  of  the  Form  x-\-  a, 
<c  +  h. 

By  actual  multiplication, 


a:  +  5 

x  —  by 

a:  +  a 

a:  +  3 

x  +  Zy 

x-^h 

a:'  +  5a: 

y}~bxy 

7?  + ax 

+  3a:  + 15 

+  3x2/- 
x'-2xy- 

-Iby^ 
-152/' 

+  hx-\-ah 

a:'  +  8a:  +  15 

a:'  +  (a  +  6)a:  +  ah. 

By  comparing  each  pair  of  binomials  with  their  product, 
the  following  relation  is  observed: 

The  'product  of  two  binomials  of  the  form  a:  +  a  and  x-\~b  con- 
sists of  three  terms : 

The  first  term  is  the  square  of  the  first  term  of  either  binomial; 
The  last  term  is  the  product  of  the  second  terms  of  the  binomials; 


ABBREVIATED  MULTIPLICATION.  ^   l^Tj 

The  middle  term  consists  of  the  first  term  of  the  binomials  with 
a  coefficient  equal  to  the  algebraic  sum  of  the  second  terms  of  the 
binomials. 

Ex.  1.    Multiply  a;  -  8  by  a:  +  7. 

-8  +  7  =  -l,        -8X7--56 
.'.(x-S){x-7)=-x'-x-c>Q,  Product. 

Ex.  2.   Multiply  (x  -  6a)  (x  —  5a). 
(-  6a)  +  (-  5a)  =  -  11a,         (-  6a)  X  (-  5a)  =  +  30a' 
.  • .  (a;  -  6a)  (a;  -  5a)  =x''-  llax  +  30a^ 

In  Case  V.,  also,  a  parenthesis  containing  two  or  more  terms 
may  be  used  instead  of  a  single  quantity.  In  all  cases  the 
student  should  thoroughly  acquire  the  ability  to  use  a  paren- 
thesis as  a  single  quantity.  By  so  doing  many  of  the  diffi- 
culties of  algebra  are  at  once  overcome. 

Ex.  3.   Multiply  x +  y-^Qhy  x  +  y —  2. 

(.x  +  y  +  Q)<ix  +  y-2)^l(x  +  y)-hQ'][(x  +  y)-2-] 

=  (.x-\-  yy  4-  A(x  +  y)  — 12,  Product. 

EXERCISE   21. 

Write  by  inspection  the  product  of — 

1.  {x  +  2)  (x  +  5).  10.  (ab  +  c)  (ab  +  3c). 

2.  (x-5)(x-S).  11.  (xy-7z')(ixy  +  ^z'). 

3.  (x  -7)(x  +  4).  ^2.  (be'  -  5a'^)  (be'  +  7a').  ^ 
®  (x  -  4)  (x  +  8).  13^"  (a;'*  -  yz)  (x'  -  4yz).      i  '' 

\Si  (x  +  l)(x-7).  14.  (xy  —  Aab)  (xy  +  ab). 

(^  (x'  -  2)  Cx'  -  3).  15.  (x"  +  7)  (a:'  -  7). 

m  (x'  +  3)  (x'  +  1).  16.  (m'  -  Sn')  (m'  -  Sn'). 


(a  +  3a:)  (a  -  lOx).  17.  (a  +  2b-  1)  (a  -  26  4-  l)o 

9.j(x  —  7y)Cx  +  y).  18.  (a- 6c  — 4)  (a  +  6c  — 4). 


So  ALGEBRA. 

19.  (a'  -  2ax  -  Zx^.  22.  (a  +  2>h)  (a  -  3c). 

20.  (a  — l)(a  +  ft).  23.  (re -f  2?/ —  7)  (:c  +  2^/ +  2). 

21.  (a  -  2x)  (a  +  y).  24.   (x^/'^'  -  10)  (xyV  +  a). 

25.  (a^6c  —  x^yz)  (cfbc  +  bx^yz). 

26.  (a6  +  ^c  —  ac)  (a6  —  6c  +  ac). 

27.  [(a +  6) -3]  [(a +  6) +  4]. 

28.  (m  —  5  +  4n)  (m  —  5  —  8n). 

91.  VI.  Product  of  Two  Binomials  whose  Correspond- 
ing Terms  are  Similar. 

By  actual  multiplication, 

2a -36 

4a +  56 
8a^  -  12a6 

+  10a6  - 156' 

W-  idb-iw 

It  is  seen  that  the  middle  term  of  this  product  may  be 
obtained  directly  from  the  two  binomials  by  taking  the 
algebraic  sum  of  the  cross  products  of  their  terms.     Thus, 

(+  2a)  (+  56)  +  (-  36)  (+  4a)  =  10a6  -  12a6  =  -  2ab. 
Hence,  in  general, 

The  product  of  any  two  binomials  of  the  given  form  consists  of 
three  terms: 

The  first  term  is  the  product  of  the  first  terms  of  the  binomials  ; 

The  third  term  is  the  product  of  the  second  terms  of  the  binomials; 

The  middle  term  is  formed  by  taking  the  algebraic  sum  of  the 
cross  products  of  the  terms  of  the  binomials. 

Ex.  Multiply  (lOx  +  7y)  (Sx  -  lly). 
For  the  middle  term, 

(lOx)  (-  lly)  +  (72/)  (Sx)  =  -  llOxy  +  56xy  =  -  5ixy. 

.  • .  (10a;  +  7y)  (Sx  -  lly)  =  80a;^  -  b4.xy  -  77y',  Product.    . 


ABBREVIATED  DIVISION.  81 

EXERCISE   22. 

Write  by  inspection  the  product  of — 
1.  (2x  +  3)  (x  +  2).  __10.  {x'  -  3x2/  +  Ay^.i^ 

%  (2x  +  b){x-  2).  n,  (5x'  -  7y)  (5x'  +  ly). 

3.  (3x  -  1)  (a;  -  2).  12.  (lUy  -  3)  O^Y  +  2). 

4.  (bx  ^l)(x-  1).  13.  (2a;'  +  ay)  (Zt?  +  2ay). 

6.  (a;  +  3^)  (32;  -  8?/).  14.  (a'  -  a  +  7)  (a'  +  a  -  7). 

6.  (2a:  -  72/)  (3a;  +  lOi/).^  15.  (x'  -  3x  +  1)  {x'  -  3x  - 1). 

7.  (3a'  -  56)  (3a'  -  56).  16.  (5x'  -  9x  -  8)1 

8.  (cr' —  22/2)  (5a;' +  3?/2).         17.  (Zx' -{- bxy  —  ^fy . 

9.  (3x  -  2/')'.  18.  (4a'  -  36)  (5a'  +  46). 

19.  [(8a;='  -{-x)-  (4a;'  -  1)]  [(8a;^  +  a;)  +  (4x'  -  1)]. 

ABBREVIATED   DIVISION. 

92.  Value  of  Abbreviated  Division.  In  certain  cases 
much  of  the  labor  of  division  may  be  saved  by  performing 
the  division  operation  in  a  typical  case,  noting  the  relation 
between  the  quantities  divided  and  the  quotient,  and  formu- 
lating this  relation  into  a  mechanical  rule.         ' 

93.  I.  Division  of  the  Difference  of  Two  Squares. 

Either  by  actual  division,  or  by  inverting  the  relation  of 
irt.  87,  we  obtain 

=  a  —  bj        and  —  =  a-\-o, 

a+b  a—b 

Hence,  in  general  language, 

The  difference  of  the  squares  of  two  quantities  is  divisible  by  the 
r,dm  of  the  quantities,  and  also  by  the  difference  of  the  quantities, 
the  quotients  in  the  respective  cases  being  the  difference  of  the  quan^ 
tities  and  the  sxim  of  the  quantities. 


82  ALGEBRA. 

Ex.  1.   -^ ^^  =  2x  +  32/,  Quotient  ^ 

2x  — 3?/ 

Ex.  2.   ^^ h   \     =  ^  -  (a  +  6),  Qwo^ten^. 

a;  H-  (a  +  6) 


EXERCISE   23. 

Write  by  inspection  the  quotient  of — 


g  (a^  +  iy-gy 

(x  +  l)  +  a 
^    a'-{h-2cf 
a-(b-2c) 

5x-6y'  '   2x'-\-(iy'  +  l) 

16a;^-4V  ^^    (a-by-(c-iy 

'     4x^  +  72/'   '  *    (a-6)  +  (c-l) 

\^    5x^-2/*   '  *    l  +  (a  +  6-c) 

94.    II.  and  III.  Division  of  Sum  or  Difference  of  Two 
Cubes. 

By  actual  division  we  can  obtain, 

^3  0-7)3  „3  _  ^3 


a-\-h  a  —  b 

Hence,  in  general  language, 

The  sum  of  the  cubes  of  two  quantities  is  divisible  by  the  sum 
of  the  quantities,  and  the  quotient  is  the  square  of  the  first  quan- 
tity, minus  the  product  of  the  two  quantities,  plus  the  square  of 
the  second  quantity ;  also, 
.    The  difference  of  the  cubes  of  two  quantities  is  divisible  by  the 


ABBREVIATED  DIVISION.  83 

difference  of  the  quantities,  and  the  quotient  is  the  square  of  the  first 
quantity,  plus  the  product  of  the  two  quantities,  plus  the  square  of 
the  second. 

Ex.  1.  §^1127/  ^  (2xy-(Sy)' 
'    '    2x  —  'iy  2x~By 

=  (2xy  +  (2x)(Zy)  +  (&yy 

=  4x^  +  6xy  +  9/,  Quotient. 

=  a'  -  2ab  +  b'-3a  +  Sb  +  9. 


EXERCISE   24. 

Write  the  quotient  of— 


a  +  2  c+(l-x) 

—.-^-1  *  2-(x  +  7/).< 

^    27x^-64  ^^^  A^-(x2/-l)» 

3a;  —  4  ic^  —  (xy  —  1) 

l+8a;«  27a;^  + 125/ 

*  l  +  2a:^*  •     3^:^  +  52/'' 

6  125-^\  ^g  (g-    iy-a;\ 

5-ar»  '  (a-l)-a;'' 

^    27a^  +  2/",  j4  Sx^  +  Ca^-iy 

*  3a=  +  2/*  '  2a;  +  ar^-l    * 

7  ^1+1!.  15  8(a:- 1/^-2^ 

*  cc'4-/  *  2(a:-2/)-z 
(g-iy-a^^  ^g  27(:r'  +  iy  +  1252" 

(a-l)-a;  '  *        3a;^  +  3  +  5«* 


84  ALGEBRA. 

95.    IV.,  v.,  and  VI.  Division  of  Sum  or  Difference  of 
any  Two  Like  Powers. 

By  actual  division  we  can  obtain, 

a*  — 6* 

=  a^  —  a?h  +  ah^  —  b^,  Quotient. 

—  a^-t-  dh  +  ah^  +  6^,  Quotient. 

But  a*  +  6*  is  not  divisible  by  either  a  +  6  or  a  —  6 : 
^'  +  ^'  =a'  -a'h  +  aW  -  ah'  +  h\ 

=  a'  +  a'b-ha'b'  +  ab'  +  b\ 


a  +  b 
a'-b' 


a  —  b 
Hence,  in  like  manner. 

The  difference  of  any  two  like  even  powers  of  two  quantities  is 
divisible  by  the  sum  of  the  quantities,  and  also  by  their  differ- 
ence ; 

The  sum  of  two  like  odd  powers  of  two  quantities  is  divisible  by 
the  sum  of  the  quantities  ; 

The  difference  of  any  two  like  odd  powers  of  two  quantities  is 
divisible  by  the  difference  of  the  quantities. 

For  the  quotient  in  all  these  cases — 

(1)  The  number  of  terms  in  a  quotient  equals  the  degree  of  the 
powers  whose  sum  or  difference  is  divided  ; 

(2)  The  terms  of  each  quotient  are  homogeneous  (since  the  expo- 
nent of  a  decreases  by  1  in  each  term,  and  that  of  h  increases  by  1 
in  each  term). 

(3)  If  the  divisor  is  a  difference,  the  signs  of  the  quotient  are  all 
plus;  if  the  divisor  is  a  sum,  the  signs  of  the  quotient  are  alter- 
nately plus  and  minus. 

The  last  statement  forms  a  general  rule. for  signs  jof  a  quo- 


ABBREVIATED  DIVISION.  85 

tient  in  all  the  cases  of  abbreviated  division,  including  I., 
II.,  and  III. 

2xi-y  2x  +  y 

=  (2xy  -  (2xfy  +  (2a;)y  -  (2x)f  4-  y' 
=  16a;*  -  ^xhj  +  AxY  -  ^xy"  +  y\  Quotient. 

EXERCISE   25. 

Write  the  quotient  of — 

^    a'-27b'  ^   x'-l 

1.    TT--  6.    -• 

x  —  1 

x'  +  l' 
81-^^ 
S  +  x"' 

x  +  y^ 

^10  _  ,  .15 

5.  ^^-^.  10.  V-r- 

Write  all  the  exact  binomial  divisors  for  each  of  the  fol- 
lowing : 

11.  1  +  Q4x\  17.  x'-y\ 

12.  dx'  -  252/*.  18-  *'°  +  2/'"- 

13.  a;* -81.  19.  l-^x-y)*. 

14.  a;^-32.  20.  x'  —  y\ 

15.  l+x\     ^  21.  a;" -64/. 

16.  x'-27y\  22.  a;'^"-^/^ 
For  what  values  of  n  \^ill — 

23.  a"  —  ^'^  be  divisible  by  a  —  6 ?     By  a  +  6? 

24.  a"  +  6"  be  divisible  bya  —  &?     Bya  +  6?     • 


a- 36 

a* -166* 

a- 26 

a^  +  1 

x  +  1 

T>-24S 

x-3 

x'  +  y' 

3.  — -•  8. 

x  +  1 

4.  ^243.  ,,_ 


CHAPTER   VIII. 
FACTORING. 

96.  The  Factors  of  an  expression  (see  Art.  12)  are  the 

qiiiintities    which,   multiplied    together,   produce   the   given 
expression. 

Factoring  is  the  process  of  separating  an  algebraic  ex- 
pression into  its  factors. 

97.  Illustration  of  Value   of  Factoring.      If  a  fraction, 

and  it  is  known  that 

and  that  2x''  -  ISx  +  21  =  (2x  -7)(x-  3), 

for  the  original  fraction  we  may  write, 

(x-B)(x-5) 
(x-Z)(2x-iy 

then  cancel  out  the  factor  x  —  3,  common  to  both  numerator 
and  denominator,  and  obtain  the  simple  fraction, 

x-b 

2x-l' 

This  is  an  illustration  of  the  usefulness  ojf"  a  knowledge  of 
factoring  in  enabling  us  to  simplify  work  and  save  labor. 

98.  A  Prime  Quantity  in  algebra  is  one  which  cannot  be 
divided  by  any  quantity  except  itself  and  unity. 

Exs.   a,  6,  a^  +  h\  17. 

86 


FACTORING.  87 

99.  Perfect  Square  and  Perfect  Cube.  When  an  expres- 
sion is  separable  into  two  equal  factors,  the  expression  is  called 
a  perfect  square,  and  each  of  the  factors  is  called  the  square  root 
of  the  expression. 

Ex.  9aV  =  ^ax'  •  3ax^ 

.  • .  Sax^  is  the  square  root  of  9aV. 

Also,  oi^  —  4x  +  4:=(x  —  2)(x—  2),  and  is  therefore  a  perfect 
square,  with  a;  —  2  for  its  square  root. 

When  an  expression  is  separable  into  three  equal  factors 
the  expression  is  called  a  perfect  cube^  and  each  of  the  factors 
is  called  its  cube  root. 

Ex.  27a'xy  =  SaxY  •  SaxY  •  Zay^if. 

. ' .  ^axY  is  the  cube  root  of  27aV2/'. 

100.  Factors  of  Monomials.  Since  monomials  are  formed 
by  simply  indicated  multiplications,  the  factors  of  a  mono- 
mial are  recognized  by  direct  inspection. 

Thus,  the  factors  of  7aV  are  7,  a,  a,  x,  x,  x. 

101.  Factors  of  Polynomials.  When  binomials  or  trino- 
mials are  multiplied  together,  simplifications  are  often  made 
in  the  product  obtained  by  the  addition  of  similar  terms. 

Hence,  given  the  simplified  result,  the  problem  of  deter- 
mining the  quantities  multiplied  (or  factors)  is  made  more 
difficult,  and  different  cases  must  be  carefully  discriminated. 

CASE  I. 

102.  A  Polynomial  all  of  whose  Terms  contain  a  Com- 
mon Factor. 

Ex.  1.   Factor  3a;'  +  6x. 

Each  term  contains  the  factor  3a;. 

Divide  3a;'  +  6a;  by  3a;,  and  the  quotient  is  a;  +  2. 

The  factors  are  the  divisor  and  quotient. 

.  • .  Sa;"  +  6a;  -=  3a;(x  +  2),  Factors, 


88  ALGEBRA. 

Ex.  2.   Factor  \2xY  -  16x/  +  ^x\j\ 

12x'y  -  16x2/*  +  ^^Y  =  4a:2/X3a:  -  4^/  +  2rc2/'). 
Hence,  in  general, 

Divide  all  the  terms  of  the  polynomial  by  the  common  factor ; 
The  factors  will  be  the  divisor  and  quotient. 

EXERCISE  26. 

Factor — 

1.  2x'-{-5x'.  4.  So:' -a.  JT^^lSx' ~ 27x'y. 

2.  7?-  2x.  6.^  la  +  Ua\  8.  x''-x^~  x\ 

3.  x"  +  X.  \  U'x"  -  15aV.  9.  a\  -  2a'x\ 

10.  3a'  -  ^ax  +  ^x\  :13,..a*6^c  -  a'6V  +  2d'h'c\ 

11.  2a:  +  4a;'  -  Qx\  14,.  2a:Y  -  8x'y  +  Gx'Y- 

12.  lQa^6'  -  35a'6l ^'J  y  15.  a"^6'c'"  +  lla"^6V"  +  \ 

CASE   11. 
103.  A  Trinomial  that  is  a  Perfect  Square. 

By  Arts.  85  and  86  a  trinomial  is  a  perfect  square  when  its 
first  and  last  terms  are  perfect  squares  and  positive,  and  the 
middle  term  is  twice  the  product  of  the  square  roots  of  the 
end  terms.  The  sign  of  the  middle  term  determines  whether 
the  square  root  of  the  trinomial  is  a  sum  or  a  difference. 

Ex.  1.    Factor  16a;'  -  2Axy  +  %'. 

This  trinomial  satisfies  the  conditions  for  being  the  square 
of  4x  —  3?/. 

.  • .  16a;'  —  2Axy  +  %'  =  (4a;  —  3^/)  (4a;  -  3^/),  Factors. 

Ex.  2.   Factor  a*  +  4a'6  +  4^'. 

a*  +  4a'6  +  46'  =  (a'  +  26)',  Factors. 

Hence,  in  general,  to  factor  a  trinomial  that  is  a  perfect  square, 

Take  the  square  roots  of  the  first  and  last  terms ^  and  connect  these 
by  the  sign  of  the  middle  term ; 

Take  the  result  as  a  factor  twice. 


FACTORING,  *  89 

^  EXERCISE   27. 

Factor — 

1.  4x^  +  ^xy  +  y\  \^.  4x^  +  44a:y  +  121a:/. 

2.  16a'^  -  24a2/  +  9/.  11.  81a^6  +  126a^6^  +  49a6'. 

3.  25x^  ~  lOx  +  1.  12.  My  -  40aa:2/  +  ^^x\j^ 

4.  a;"'  -  20a;2/  +  1002/1  13.  2a;*  ~  8x'  +  8x^ 
J|f=  49c  +  286c'^  +  Wc\  <4^0^^2/  +  Bx*  +  752/^ 

^6.  d'W-  -  6a6c  +  9cl  15.  a^x  +  aa;'  -  2aV. 

j^  a;?/'  +  2x2/  +  ^-  16.  rc'^"  +  Ix^'y  +  2/'- 

8.  2m='n  —  4m7i  +  271.  17.  {a-hj  —  Icia-h)  ^  &. 

9.  a^  +  2a*  +  al  18.  9(a;  +  yj  +  122(a;  +  2/)  +  ^  °^C 

19.  16(2a  -  3)' -  16a6  +  246  +  6^  "^ 

20.  2o(a;  -  yj  -  \%)xy{x  -y)  +  144a;Y. 

21.  a^  +  6^ +'c2  +  2a6  +  2ac  +  26c. 

CASE    III. 
104.  The  Difference  of  Two  Perfect  Squares. 

From  Art.  87,      (a  +  6)  (a- 6) -a'- 6^ 
hence,  a^  —  6^  =  (a  +  6)  (a  —  6).  ^ 

But  any  algebraic  quantities  may  be  used  instead  of  a  and 
h.    Hence, 

Ex.  1.   Factor  x^  —  l^y\ 

x^  —  IQy'^  =  (a;  +  4y)  (x  —  Ay)^  Factors. 
In  general,  to  factor  the  difference  of  two  squares, 
Take  the  square  root  of  each  square  ; 
The  factors  will  be  the  sum  of  these  roots  and  their  difference. 

Ex.2.   x'-y'  =  (x'-hy')(x'-y') 

=  (ar*  +  2/0  (a;  +  y)  (x  —y),  Factors. 

*  Apply  Case  I.  first. 


90  *  ALGEBRA. 

105.  Special  Cases  under  Case  III.  A.  We  may  also 
factor  by  this  case  the  difference  of  two  squares  when  one 
or  both  of  the  given  squares  is  a  compound  expression. 

Ex.  1.   Factor  (a  +  2by  -  4x\ 

(a  +  2by  -  Ax'  =  [(a  +  26)  +  2a;]  [(a  +  26)  -  2a;] 

=  (a  4-  26  +  2a;)  (a  +  26  —  2a:),  Factors. 

Ex.  2.    Factor  (3a;  +  4yy  -  (2a;  +  Syy. 
(3x  +  4yy  -  {2x  +  Syy  =  [(3a;  +  4y)  +  (2a;  +  37/)]  [(3a;  +  4y)  -  (2x 

+  32/)] 
=  (3a;  +  4y  +  2x  +  Zy)  (3a;  +  4y-2x-  Zy) 
=  (6a;  +  7y)  (x  +  2/)>  Factors. 


Factor — 

EXERCISE   28. 

1.  x'-9. 

2.  25-16a^ 

8.  a;y-362/. 
9.*  x^  —  a;. 

15.  a«-4a;*. 

16.  2a'6*-98a6. 

3.  4a' -4961 

10.  a;«-25. 

17.  144 -a;^'. 

4.  Sx'-12y\ 

5.  100 -81m'. 
G.  m  —  Mmn\ 

^3a;^-75x/. 

12.  ,^2 -2a;*. 
]^V-a:2/*. 

14.  a* -816*. 

18.  x^-\mf\ 

\^  x^-y\ 
""20.*.,16aa;*-81a. 

7^49a:*-l. 

21.  225a;'" -y'.     ' 

22.  x^  -  /.                                  29.  (x  +  2yy  -  (3a;  +  1)'. 

23.  x"-  -  f-z\                           30.  25(2a  -  6)'  -  (a  -  36)'. 

24.  {x  +  yy  -  1.                        31.*  a;^y  -  yz^\ 
@.  a;'  -  (3/  +  1)'.          .             32.  81a;^^  -  16/. 

26.  {x  -  yy  -  9.                        33.  x"^  -  144a;/2^ 

27.  4(a;  -  yy  -  25.                    34.  {a  -  by  -  4(c  +  1)1 

28.  l-36(a;  +  22/)'.                  35.  1  -  100(a;' -  a;  -  1)'. 

*  May  be  resolved  into  four  factors. 


FACTORING,  '91 

106.  B.  Grouping  of  Terms.  It  may  be  possible  to  group 
or  rearrange  the  terms  of  a  polynomial  expression  so  as  to 
produce  the  difierence  of  two  squares. 

Ex.  1.    Factor  x^  —  4xy  +  4y'  —  9z\ 
x"  -  4x2/  +  42/'  -  ^z'  =  (x'  -  4xy  +  4y')  -  9^' 
=  (x~2yy-9z' 
=  [(x-2y)  +  Sz-]l(x-2y)-Bz'] 
=  (a;  —  22/  +  3z)  {x  —  2y~  3z),  Factors, 

Ex.  2.   Factor  2xy  —  x' +  a' -  y\ 

2xy  -x'  +  a'-y'  =.a'  -  (x'  -  2xy  +  f) 
=^a:^-(x-yy 
\  =  [«  +  (^  -  2/)]  [a  -  (a;  -  :y)] 
=  (a  +  a:  «^  2/)  (<^  ~  ^  +  3/)>  Factors. 

Ex.  3.   Factor  a'  -  ic'^  -  2/'  +  ^'  +  2a6  +  2xy. 
a»  -  ic^  -  2/'  +  6'  +  2a6  +  2a;2/  =  {a'  +  2a6  +  h')  -  {x^  -  2xy  +  y') 

=  (a  +  by-{x-yy 
=  (a-^b  +  x  —  y)(a  +  b  —  x  +  y'), 

Factors. 


Factor— 

1.  x'-ie(x-2yy. 

2.  9(a-6y-25. 
8.  a'~2ah-{-b'-l. 

4.  9x''  +  12xy^-4:y'-z\ 

5.  x'-^a'-y'  -2ax 


6. 

a' 

-^f- 

-  a:'  +  2ay. 

7. 

a' 

-a:'- 

-2x'y-y\ 

8. 

x" 

-f- 

-1-21/. 

EXERCISE 

:  29 

• 

9. 

l  +  2xy 

-^-3/'. 

10. 

e-a^- 

-6'  +  2a&./l 

11. 

a'  +  b'- 

-c^-2a6r^ 

■z\ 

12. 

4.~x'- 

4y^  —  4x2/. 

13. 

2a +  b'- 

-  a'  -  1. 

14. 

2ab  +  a'. 

b'  +  l-x". 

15. 

2z'  -  4z 

-  2z*  +  2. 

16. 

9x'^  +  2/' 

-25z^-6a^. 

92  ALGEBRA. 

17.  2O2/Z  +  a;' —  42/' —  25z^ 

18.  45x'  —  20a;*  -  bf  —  20icy 

19.  a^  +  2ah  -\- b' -  c' -  2cd  -  d\ 

20.  af  +  4^/'  -  92'  -  1  -  Axy  -  6z. 

21.  ^a^  -  25a;'  +  46'  -  1  -  10a;  -  12a6. 

22.  a'  -  96V  -  1  +  66a;  -  10a6  -f  2561 

23.  16a;^  -  lQ>x'y  -  16x  -  48a:z  -  36a;z'  +  Axy\ 

107.  C.  The  Addition  and  Subtraction  of  Some  Quan- 
tity will  sometimes  transform  an  expression  into  a  difference 
of  two  perfect  squares. 

Ex.  1.   a*  +  a'6'  +  6*  =  a*  +  2a'6'  +  6*  -  a'6' 
=  (a'  +  6')'-a'6' 
=  (a'  4-  6'  +  ah)  (a'  +  6'  —  a6),  i^actors. 

Ex.  2.   a;*  —  7a;'2/'  +  2/*. 
Add  and  subtract  9a;y. 

ic*  —  IxY  +  2/*  =  a;*  +  2a:y  +  2/*  —  9a;y 

=:(a;'  +  2/^)^-9xy 

=  (a)'  +  2/'  +  3j2/)  (a:'  +  2/'  "  3^2/),  i^actors. 

EXERCISE   30. 

Factor — 

1.  c*  +  cV  +  a;*.  7.  49a;*  +  34a;V  +  252*. 

2.  a;*  +  a;'  +  l.  8.  16a;*- 9a;' +  1. 

3.  4a;*  -  13a;'  +  1.  9.  100a;*  -  61a;^  +  9. 

4.  4a*  -  21a'6'  +  96*.  10.  100a;*  +  11a;'  +  9. 

6.  9a;*  +  3a;'2/' +  42/*.  11.  225a*6*  -  4a'6' +  4. 

6.  49c*  ~  We'd'  +  2bd\  12.  32a*  +  26*  -  56a'6'. 

13.  a*  +  46*.  14.  1  +  64a;*.  15.  xSf  +  324. 


FAGTOBINQ.  93 

EXERCISE   31. 

SPECIAL  REVIEW. 


Factor- 


1.  1  -  4a:  +  ^x\    /  -^  7j  (/'!^  1<^-  2«^*  -  2a5. 

2.  12^?/2  -  3x'\     y  ^n,  ^V.i'V^  '   fef49  -  140712  +  iqOti*. 

3.  9  +  a;2  -  6a;  -  )/^.  ,v^^  *I2:  4^2  +  a;*  -  4a;='  -  1.  W 
, ,  ,  ^v  a;  -  2a:»  +  a;5.     v.-_\T  -^.^S^-^^'-'^S^      -  ISa;^  +  a;*: 

6.  a:*  -  14^-2  ^  i  '^^^'  ^a:^  -  3  -  3a*  +  6^ 

7.  20a:V  -  45^^  ^^    16    ^^2  +  ^^2^2  _  4^2yj2^ 

8.  fa^  _  52  _  ^2  +  26c.^  ^.  .;,  -N    „.    17.  4a:y  -  (a;^  +  y""  -  \)\ 

9.  32a:5  -  2a;3  +  2a;.     , ,  ', ..  /V.^  18.  ^a'x'  +  256^  _  20a6a;.i 

19.  2a:^*^  ^xy"^  -  2a;  +  4a;y. 

20.  1  -  6a6  -  952  +  I2a25  _  4^*  +  9^252^ 

21.  a'x'^  -b^  -y^  +  1-  2ax  +  2by. 

22.  9aW  -  25  -  IGa^o^s  +  4^2  -  I2a6a;  -  40aa;. 

23.  (a;2  +  ^2  _  9)2  _  4^222^  24..  49a;*  +  66a;2^*  +  25^8. 

CASE   IV. 

108.  A  Trinomial  of  the  Form  x^  +  6a?  -f  c. 

It  was  found  in  Art.  90  that  on  multiplying  two  binomials 
lixe  a;  +  3  and  a;  —  5,  the  product,  x^  —  2x  — 15,  was  formed  by 
taking  the  algebraic  sum  of  +  3  and  —  5  to  obtain  the  coeffi- 
cient of  X, — viz.  —  2, — and  taking  their  product,  — 15,  to  form 
the  last  term  of  the  result.  Hence,  in  undoing  this  work  to 
find  the  factors  of  x^  —  2x  —  15,  the  essential  part  of  the  pro- 
cess is  to  find  two  numbers  which,  added  together,  will  give 
—  2,  and  multiplied  together  will  give  —  15. 

Ex.  1.   Factor  x'  +  llx  +  30. 

The  pairs  of  numbers  whose  product  is  30  are,  30  and  1,  15  and  2,  10 
and  3,  6  and  5.     Of  these,  that  pair  wjtiose  sum  is  also  11  is  6  and  5. 

Hence,  x^  +  11a;  +  30  =  (a;  +  6)  (a;  +  5),  Factors. 


94  ALGEBRA.  * 

Ex.  2.   Factor  x"  -  8fc  +  7.  ^       K 

It  is  necessary  to  find  two  numbers  whose  product  is  +7,  and  sum  is  —8. 

When  the  last  term  is  positive,  as  in  this  example,  the  two  required  num- 
bers must  be  both  positive  or  both  negative,  and  since  their  sum  is  negative, 
they  must  be  both  negative. 

. ' .  x^  -  8x  +  7  =  {x  -  7)  {x  -  1),  Factors. 

Ex.  3.   Factor  a;'  —  a;  —  30. 

It  is  necessary  to  find  two  numbers  whose  product  is  —30,  and  sum 
is  -1. 

Since  the  sign  of  the  last  term  is  minus,  the  two  numbers  must  be  one 
positive,  the  other  negative  ;  and  since  their  sum  is  —  1,  the  greater  number 
must  be  negative.  >,  ..^ 

x'  -  X-  SO  =  (x-  6)  {x  +  5),  Factors.  ^^ 

Ex.4.   Factor  cc' +  3x3/ -  lOy. 

Since  5^,  —  2^,  added  give  3^,  multiplied  give  — 10^', 

x^  +  Sxy  -  10^2  =  (a;  +  5^)  {x  -  1y),  Factors. 

Hence,  in  general,  to  factor  a  trinomial  of  the  form  a? 
H-  hx  4-  c, 

Find  two  numbers  whichy  multiplied  together,  produce  the  third 
term  of  the  tnnomial,  and  added  together  give  the  coefficient  of  the 
second  term; 

X  (or  whatever  takes  the  place  of  x),  plus  the  one  number ^  and  x 
plus  the  other  number ^  are  the  factors  required. 


EXERCISE   32. 

Factor — 

1.  x'  +  5a;  +  6.  6.  x'  +  x- 30. 

2.  a;'  —  a;  -  6.  7.  x'  +  Qxy  -  IG/. 

3.  a:'  +  a;  -  6.  8.  a:'  -  Qxy  -IQf^ 

4.  a:'  4-  7x  —  44.  9.  x'  +  8a;  -f^l6.* 

5.  X*  -  Ux  +  30.  10.  a;'  +  5x  -  36. 


FACTORING.  95 

11.  x^-hx-  36.  C^)ia;^  ~  9a;'  +  8.  s.^ 

12.  a;*  -  bx^  -  36.  25.  2a  -  14aa;  -  60aa:'. 

13.  x^  +  Zx-  28.  26.  *>:e''  -  22a:'  -  120a:. 

14.  a;'  -  2a;  -  48.  27.  ■  25a;y  +  b^y  -  3(^2/*. 

15.  a:'  -  8a;  -  48.  28.  56«a;'2/  +  96ax2/  +  ^a7?y. 

16.  a;'  +  16a;  +  48.  ^29.  2a;«  -  10a;'  -  28a;'. 

17.  x'  +  19a;  +  48.  J^^  +  ^-  20a;. 

18.  a;'  +  13a;  -  48.  31.  r'  -  25x'  +  144x. 

19.  x'  -  22x  -  48.  82.  3a;»  -  51a;*  +  48.. 

20.  a;'-49x  +  48.  33.  a;'" -a;" -56. 

21.  ar^  -  4a;  -  96.  34.  aW  -  llaic'  -  26c*. 
Jg^  xY  -  23xy  +  132.                  35^  Zaxf  -  9ax'y  -  SOas^. 

^x"  -  dax  -  24a\  36.  5a;'  +  SOs^y'  -  35x3/*.  . 

37.  x'-{-(a  +  b)xfab. 

38.  x"  +  (2a  -  36)a;  -  6a6. 

39.  a;'  -  (a  +  26')a;  +  2a6'. 

40.  a;'  +  (a  +  26  +  c)x  +  (a  +  6)  (6  +  c).  n^. 
i\:  7^  +  (a  +  h)x  +  (a  -  c)  (6  +  c). 

42.  (a;-2/)'-3(a;-2/)-18. 

43.  2(a;'  +  2x)'  -  14(ar^  +  2x)  - 16.^ 

CASE  V. 

109.  Trinomial  of  the  Form  ax^  +  hx-\-c. 

From  Art.  91  it  is  evident  that  the  essential  part  of  the 
process  of  factoring  a  trinomial  of  the  form  qt?  -{-hx-\-  c  lies 
in  determining  two  factors  of  the  first  term  and  two  factors 
of  the  last  term,  such  that  the  algebraic  sum  of  the  cross 
products  of  these  factors  equals  the  middle  term  of  the  tri- 
nomial. 

Ex.  Factor  10a;'  +  13x  -  3. 


96  ALGEBRA. 

The  possible  factors  of  the  first  term  are  lOx,  x ;  and  5x,  2a^ 
The  possible  factors  of  the  third  term  are  —  3,  1 ;  and  3,-1. 
In  order  to  determine  which  of  these  pairs  taken  together 
have  the  sum  of  their  cross  products  equal  to  +  13x,  it  is 
convenient  to  arrange  the  pairs  thus : 

10a:,  -  3  5a;,  -  3 


X,      1  2x,      1 

Variations  of  these  may  be  made  mentally  by  transferring 
the  minus  sign  from  3  to  1 ;  and  also  by  causing  the  3  and 
the  1  to  change  places. 

It  is  found  that  the  sum  of-  the  cross  products  of 
5a;, -1     .         _ 

2.;  3  ^^+^^^- 

Hence,        lOa;'^  +  13a;  -  3  =  (5a;  -  1)  (2a;  +  3),  Factors. 

Hence,  in  general,  to  factor  a  trinomial  of  the  form  ax^ 
+  bx  +  c, 

Separate  the  first  term  into  two  such  factors,  and  the  third  term 
into  two  such  factors,  that  the  sum  of  their  cross  products  equals  the 
middle  term  of  the  trinomial ; 

As  arranged  for  cross  multiplication,  the  upper  pair  taken  to- 
gether and  the  lower  pair  taken  together  form  the  two  factors. 


3tor — 

EXERCISE   33. 

ft.  2x^  +  33;  4- 1. 

7.  6a;^  +  20a;'  -  16x 

2.  3a;'^-14a;  +  8.  ^ 

8.  3a;' -4a; -4. 

3.,  2a;'^  +  5a;  +  2. 

9.  8a;'  +  2x~15., 

4.  3a;'  +  10a;  +  3. 

10.  2a;'  +  x-m 

5.  6a;' -  7a;  —  5/^' 

11.  12x'-5a;-2. 

6.  23^^  +  53; -3. 

12.  4a;'  +  llx-3, 

FACTORING,  97 


"•^^^5a;'  +  24x-5.  22.  12a:'-7a;2- 122». 

9a:'  -  Ibx'  -  6x.  23.  24x'  +  104xy  -  18ajy*. 


16.  6x^2/  -  2x2/  -  42/.  ^A  ^  24.  25a*  +  9a'6' 

(^iTl 6a^-J6xJ^- 272^^  25.  IGx^-lO^y 


- 166*. 
cy-92/*. 

17.  122;'^  +  iC2/ — ^3p^  26.  3a;'"  -  8x"2/ -  82/'. 

18.  422:'  +  13x  -  42.  27.  25a*  -  41a'^6'  +  166*. 

19.  32a* +  4a6- 4561  28.  36a;*  -  97xy  +  361/*. 
420.  4x*  -  ISx'^  +  ^.4  29.  20  -  9x  -  203:^. 

21.  9x*  -  148a:^  +  64.  30.  5  +  32a:y  -  21af^^.^ 

31.  (a  +  6)' +  5(a  +  6)  -  24. 

32.  3(x-2/)'  +  7(a;-2/>-6z^ 

33.  3(a;*  +  2xf  -  5(x^  +  2a;)  -  12. 

34.  4a:(a:'  +  3a;)'  -  d>x(o^  +  3x)  -  32a;. 

35.  2(a;  +  l)'-5(a;'-l)-3(a;-l)».- 

CASE  VI. 
110.  Sum  or  Difference  of  Two  Cubes. 

From  Art.  94,        .^^+Z  =a'-ab  +  h\ 
a  +  b 

Hence,  a' +  6»  =  (a  + 6)  (a' -a6 +  6')  ....  (1) 

In  like  manner,  a'  — 6'  =  (a  — 6)  (a' +  a6  +  6')  ....  (2) 

But  any  algebraic  expressions  may  be  used  instead  of  a  and 
6  in  (1)  and  (2). 

Ex.  1.   Factor  27a;'  -  Sy". 

2l7?-S^  =  (Zxf-(2y)\ 
Use  3a;  for  a  and  2y  for  6  in  (2)  above. 

273;^  -  82/'  =  (3a;  -  2y)  {^x^  +  Qxy  +  42/'),  Factwi. 

Ex.2.   Factor  a«  +  86'. 

a«  +  86'  =  (a')'4-(26»)» 

=  (a'  +  26')(a*-2aV  +  460, 
7 


y 


98  ALGEBRA. 

Ex.  3.   Factor  (a  +  hf  -  a:', 
(a  +  6)'  -  a^  =  [(a  +  6)  -  a:]  [(a  +  h)'  +  (a  +  6)a;  +  a^]. 

Hence,  in  general,  to  factor  the  sum  or  difference  of  two 
cubes, 

Obtain  the  values  of  a  and  h  in  the  given  example^  and  substitute 
these  values  in  either  the  formula  (1)  or  (2). 

111.  Sum  or  Difference  of  any  Two  like  Odd  Powers. 

Since  the  difference  of  two  like  odd  powers  is  always  divisible 
by  the  difference  of  their  roots  (see  Art.  95),  the  factors  of  a" 
—  6",  when  n  is  odd,  are  the  divisor,  a  —  b,  and  the  quotient. 

Ex.  1.   a'-b'  =  {a-  b)  (a*  +  a'b  +  a'b'  +  ab'  +  b'). 

Since  the  sum  of  two  like  odd  powers  is  divisible  by  the  sum 
of  the  roots  (see  Art.  95),  the  factors  of  a"  +  6**,  when  n  is  odd, 
are  the  divisor,  a  +  6,  and  the  quotient. 

Ex.2.   a^  +  Z2^  =  x'-^(2yy. 

=  (ix  +  2y)  [x'  -  x\2y)  +  x\2yy  -  x(2yy  +  (2?/)^ 
-={x  +  2y)  [x*  -  2x'y  +  4xy  -  Sxy'  +  IQy'l  Factors. 

112.  Sum  or  Difference  of  any  Two  Even  Powers. 
The  difference  of  two  even  powers  is  factored  to  best  advan- 
tage by  Case  III. 

Ex.1.   x'-y'  =  (x'-{-y')(x*-y*) 

=  (a;*  +  2/*)(^  +  2/')(a^-yO 

=  (x'  -\-y')  (i^  +  y')(x  +  y)  (x-y). 

The  sum  of  two  even  powers  cannot  in  general  be  factored 
by  elementary  methods  unless  the  expression  may  be  re- 
garded as  the  sum  or  difference  of  two  cubes  (Art.  110),  or 
other  like  odd  powers. 

Ex.2.   a^  +  b'  =  (ay  +  (by 

=  (a'  +  b')  (a*  -  a'b'  -h  b*),  Factors. 

But  a'  +  6',  a*  +  6*,  a®  +  b^,  cannot  be  factored  by  any  ele- 
mentary method;  and  are  therefore  prime  expressions. 


FACTORING.  99 

EXERCISE    34. 

Factor — 

1,  m^—n\  8.  1  — 1000a;3.  15.  250a;~2a;'^. 

2.  c^  +  Sd\             9.  21x^-\-a^x.  16.  8^«  +  ^/^ 
3.27  —  ^.             10.  512a:3  — ^6.  17.  («+ ^,)3^  1 

4    a?-\-mcK         11.  a  +  343a*.  18.  125  +  (2&  — a)3. 

5.  a^3  — 125.  12.*  a6  — ^6^  19.  s—(c  +  d)\ 

6.  642/3  —  27.         13.  a;i2_^6  20.  {x—y)^—21x^. 

7.  a^fes  +  l.  14.  a6_ 64^12  2I.  16xY—5^xz\ 


22.  ar5  +  i/5  27.  aii  +  a:ii.  32.  64— (a  — 6)3. 

23.  x'^—y\  28.  a^ -f  ^,9  33  8(a;  — 22/)3  +  l 
24. t  rt^  +  m6.  29.  32a;5— 1.  34.  a^^  —  b^\ 

25.  2^12  ^  yi2  30.  ^11  _  2,11.  35  ^10  _{_  JIO 

26.  a^— 1286^  31.  243  — x^.  36.  S2x^~a^\ 


CASE  VII. 
118.  A  Polynomial  whose  Terms  may  be  grouped  bo 
as  to  "be  Divisible  by  a  Binomial  Divisor. 

Ex.1,    ax  —  ay — hx-\-hy  =  {ax  —  ay)  —  (bx  —  by) 
=  a{x  —  y)—h{x  —  y) 
=  {a—h)  (x  —  y) ,  Factors. 

Ex.  2.     l-|-15a*--5a  — 3a'  =  l  — 3a'  — 5a  +  15a* 

X    ={l  —  3a')-5a{l  —  3a') 
={l  —  3a^)  (1  — 5a),  Factors. 

Ex.3.     «'+2/'+ar  +  2/  =  (a;  +  y)  (x' —  xy -[- y') -{- {x  +  y) 

=^{x+y)  {o(^  —  xy-}-y^-^l),  Factors. 

Ex.  4.    o'  4-  8a'  — 4  =a'  +  2a2  +  «'  —  4. 

=  a2(aH-2)  +  (a  +  2)  (a  — 2) 

=  (a  +  2)  (a'  +  «  — 2) 

=  (a  +  2)  (a  +  2)  (a  — 1),  Zacfor*. 

*U8e  Case  III  first.  tSum  of  two  cubes. 


100  ALGEBRA, 

EXERCISE   35. 

Factor — 

1.  ax -^  ay -\- hx -\-  by.  11.  x^  -\-Sy  —  Sx  —  xy, 

2.  x^  —  ax  +  cx  —  ac.  12.  z'  —  z^  —  z  +  1. 

3.  5xy  —  lOy  —  Sx  +  Q.  13.  ab  —  by  —  a  +  y. 

4.  3am  —  4mn  —  Qay  +  Sny.  14.  x^  —  «*  —  4x  +  4. 

6.  a'x  +  Sax  +  aca;  +  Sex.  15.  aV  -  6'a:'  -  ay  +  by. 

6.  3a''2/  +  ^(^h  —  5«^2/  -  ^^^2/-     16.  a;(x  -f  4)'  4-  4(:r  +  4). 

7.  a;*  +  x*  +  2a;'  -f  2x.  '  17.  a^a  +  3)  -  3(a  +  3). 

8.  2x*  -  2a;'  -  2aV  +  2a'a;.         18.  2(a;'  -  y')  -(x-  y). 

9.  2/'  4-  2/'  +  2/  +  1-  19.  4a;(a)  -  l)'^  +  x  —  l. 
10.  ox'  —  2a'a;  —  a;  +  2a.                20.  a;'  —  1  +  2(a;'  —  1). 

^\  21.  4a'-aV+a;'^-4. 

22.  Aa3i^  +  Sax  —  Sa  —  4aa^. 

23.  a(3a-a;y-6ax'  +  2a;'. 

24.  a:'-8-7(a:-2). 

25.  4(a;'  +  27)-31a:-93. 

26.  (2a: +  iy- (2x4-1)  (3a; +  4). 

27.  (2a;  -  3)»  +  2a;' -  9x  +  9. 

28.  a;»  — 7a;-6. 

29.  a;»  —  3a;'  -  10a;  +  24. 

30.  a;'  -  8a;'  +  17a;  - 10. 

31.  6a;' -  23x' +  16a;  -  3. 

114.  General  Principles  in  Factoring.  Ii>  order  that  the 
application  of  factoring  may  be  as  effective  as  possible,  it  is 
important  to  reduce  each  expression  factored  to  its  prime  fac- 
tors. Hence  it  is  important  to  use  the  different  methods  of 
factoring  in  such  a  way  as  to  give  prime  factors  as  a  result 
most  readily. 

Hence,  in  factoring  any  given  expression,  it  is  useful  to — 


FACTORING.  101 

1.  Observe,  first  of  all,  whether  all  the  terms  of  the  expres- 
sion have  a  common  factor  (Case  I.) ;  if  so,  remove  it. 

2.  Determine  which  other  case  in  Factoring  can  be  used 
next  to  the  best  advantage. 

8.  If  the  expression  comes  under  no  case  directly,  try  to 
discover  its  factors  by  rearranging  its  terms,  or  by  adding 
and  subtracting  the  same  quantity  to  the  given  expression, 
or  by  separating  one  term  into  two  terms. 

4.  Continue  the  process  of  factoring  till  each  factor  can  be 
resolved  no  further. 

Ex.  1.   Factor  x'  —  dx\ 

This  expression  as  it  stands  might  be  factored  as  the  difiference  of  two 
squares  (Case  III.),  but  it  is  best  to  apply  Case  I.  first. 

X^  -9x''-=  x\x^  -  9)  - 

=  x\x  +  3)  (a;  -  3),  Factors. 

Ex.2.   Factor  a«-6^ 

This  expression  might  be  factored  by  dividing  it  by  a  +  6  or  by  a  —  6, 
and  taking  the  divisor  and  quotient  as  the  factors,  but  it  is  factored  to  best 
advantage  by  the  use  of  Case  III.,  and  afterward  Case  VI. 

a«  -  6«  =  (a'  +  6»)  (a^  -  h^) 

=  (a  +  6)  (a'  -  ab  +  b^)  (a  -  6)  (a^  +  a6  +  6'),  Facton, 

Ex.  3.   Factor  a;'  -  Qx'  +  6a;  -  5. 

7?  -Qx"^  -^  Qx-b==Q^  -Qx"^  ^bx  +  x-h 

^x{x-l){x-b)^{x-b) 

=  (x  -  5)  [x{x  -  1)  +  1] 

=  {x  -  b)  {x"^  -  X -\-  1),  Factori, 

Ex.  4.   Factor  Gr'i/  —  Sx^y  —  Ixf  -  \xY  +  2x2/. 
^7?y  -  ^x'^y  -  2xy^  -  ^x^y"^  +  2xy 

=  2xy{Zx^  -  Ax-y"^  -'Ixy  +  1) 

=  1xy{Ax^  -  Ax  -V\  —  x^  -  Ixy  —  y*) 

=  2a:^[(2a:  -  1)'^  -  {x-^yf-^ 

=  2xy{^x-\-y  -1)  {x—y  -  1),  Facton, 


102 


ALGEBRA. 


■  l 


Ffictor— 
1.  Sx^  —  Sx. 

%JLx'  —  't^x'y  -f  '^xf, 
8.  a;^-lla:  +  30. 
4.  \^  +  hxy  —  62/'. 
^12a'  -  lah  -  "m^J 
U  a:*- 1-2/^  +  22// 
T'40a'  -  5. 
^16x^-40^2/  +  252/^ 
l^^'^  +  Zax  —  3a  -  x. 
10.  3x'-3a:. 
l^g*  -  5a'  +  1. 
j|*2x«-32. 
^./  +  4a;  —  45. 
14.  4a;"'  +  2a  -  a'  -  1. 
'  ^"^IS^-  5ax^  —  5a.  - 
':i-  liJ-  18x'  -  3a:'  -  36x. 
«*  +  3xV  +  4z*. 

18.  oV-9a;'-a'  +  9^ 

19.  110 -a: -a:'. 

20.:;  3x'  +  13x7/  -  3O2/'. 
'^;  7a  -  lan)\ 
|22.;6x'  +  14a^  +  8. 

23.  a:*  -  {x  -  2)'. 

24.  3a  +  3a*. 

25.  a'-a'  +  2a-2. 
^.  6x^-2x-4a;'. 
:^27.  1  -  23z'  +  z*. 

28.  128-22/'. 


EXERCrSE   36. 


(gj  1  -  «'  -  6'  -  2a6. 

,30.  21a' -17a -30. 

31.  a;''  +  2/''. 

32.  8a:^  +  7292». 
38.  405jy-45a:*. 

/'Spa'  -  4a='  +  bd'  -  20. 

35.'  (c  +  (^)»-l. 

36.  {x-yy-^2(x-y). 

37.  24x' +  5x2/ -  36?/'. 

38.  x'  -  2x\j  -  4x2/  +  8/. 

39.  (a'-Qy~-n\ 

40.  z*  +  2'+  1. 

41.  (a'--^;'^-c7-46V. 

42.  21a;'-40x2/  — 2I2/'. 

43.  32  f  n^ 

44.  5a;' +  5x2/'. 

45.  m'  +  n\ 

46.  2ax='-f  ia2/-l 

47.  1  -f  a;  —  a;*  -  x\ 

48.  a;'-9-7(x-3)^ 

49.  4a*  -  37a'  +  9. 

50.  x'  -  64. 

61.  x'  -  27  -  7(x  -  3). 

52.  S2x^y-yz'\ 

53.  (x'  +  2/"0*-16xy. 
64.  x'  +  xY-yV  —  z\ 
66.  aa;*  —  ax  —  x^y  -\-  y. 
66.  4(a'-60-3(a  +  6)l 


FACTORING.  103 

67.  a''  - 1.  59.  4a'  -  96'  - 1  -  66. 

68.  4a'  -  96'  +  4a  -  66.  60.  Z^t?  +  18x'  -  40«. 

61.  (a;'  ~  1)'  +  (2x  +  3)  (a;  - 1)'. 

62.  a'-6*-aV  +  6V. 

63.  3a;'  -  27  +  ax'  -  9a. 

64.  18a'6  +  86  -  27a'c  -  12(;. 

65.  3x' -  3a;  +  4a;*  -  4a;'. 

^'  66.  l-4a'6'c'-9a:'2/'z'  +  12a6ca:2/2. 

67.  a'6ca;  —  amnpx  +  m^wpy  —  abcmy, 

68.  4a;  +  4an  +  a;'  — 4a'-n'  +  4. 
I                 69.  2(ia^  -  8)  +  7a;'  -  17a;  +  6. 

^  70.  a* -46*  + a' +  26'. 

71.  4a;»  —  19a;  +  15. 

72.  3a;'»  +  7a;'-4. 


73.  49a;'  -  70a;  +  25.  83.  45a;'  +  Sxy-  21y', 

74.  2/'  +  4a;  -  1  —  4a;'.  84.  aa;'  +  5ax  —  84a. 

75.  49a;*  -  22'a;'2/'  +  %*.  85.  ISar'  -  5a;'  +  33x  -  11. 

76.  5xV  -  5a;2/*.  86.  a;'t/  -  lOaj'i/'z'  +  25xy'z\ 

77.  a;*  +  ar* -a; -  1.  87.  a;* - 79a;'  +  1. 

78.  21a;'  4  2a;  -  55.  88.  a'  -  9  +  96'  -  6a6. 

79.  18a;'  +  62xy  -  Qy\  89.  a;'^  -  4a;*  -  lea;'  +  64. 

80.  (a;  +  D'  -  x\  90.  (x'  +  3)'  -  64x*. 

81.  (1  -  2x)'  -  x\  91.  «*  -  492/'  +  ^-  6«'- 

82.  ax'  -cx  +  ax-c.  92.  60a;'  +  119x  -  60. 

93.  xY  —  4x'  +  4  -  2/'  -  4xY  +  4xy. 

94.  a'nx  —  bcrri'yz  +  acmxz  -  ahmny, 

95.  5(a;^  +  27)  -  liar*  -  46x  -  -  39. 


CHAPTER    IX. 

HIGHEST  COMMON  FACTOR  AND  LOWEST  COM- 
MON MULTIPLE. 

115.  Value  of  Highest  Common  Factor  and  Lowest 
Common  Multiple.  In  the  use  of  factors  it  is  frequently 
important,  in  order  to  do  required  work  most  effectively  and 
with  least  labor,  to  be  able  to  find  the  factor  of  highest  degree 
common  to  a  number  of  given  expressions,  or  to  determine 
the  expression  of  lowest  degree  which  will  contain  exactly  a 
number  of  given  expressions. 

116.  A  Common  Factor  of  two  or  more  algebraic  expres- 
sions is  an  expression  which  divides  each  of  the  given  ex- 
pressions without  a  remainder. 

The  Highest  Common  Factor  of  two  or  more  algebraic 
expressions  is  the  product  of  all  their  prime  common  factors. 

This  product  will  evidently  be  the  factor  highest  in  degree 
that  will  divide  each  of  the  original  expressions  without  a 
remainder. 

Ex.  1.   The  H.  C.  F.  of  4x\  12a;',  16:x^y  is  ix". 

Ex.  2.   The  H.  C.  F,  of  6xXx  -  y)\  Wxi^  -  y')  is  Sx(x  -  y\ 

CASE   L 

When  the  Highest  Common  Factor  may  be  found  di- 
rectly by  Inspection. 

117.  H.  C.  F.  of  Monomials. 

Ex.  Find  H.  C.  F.  of  60aV,  45ax\  OOaV^/. 

By  arithmetic  the  H.  C,  F,  of  the  coefficients,  60, 45, 90,  is  15. 
104 


HIGHEST  COMMON  FACTOR.  105 

a  is  common  to  all  of  the  given  expressions,  and  its  least 
exponent  in  any  of  them  is  1. 

X  is  common  to  all  the  expressions,  and  its  least  exponent 
in  any  of  them  is  2. 

.  • .  15ax'  is  the  H.  C.  F. 

In  general, 

Take  the  highest  common  factor  of  the  coefficients  ; 
Annex  the  letters  common  to  all  of  the  expressions^  giving  to  each 
letter  the  least  exponent  which  it  has  in  any  expression. 

118.  H.  C.  P.  of  Polynomials  directly  Factorable. 
Ex.  1.  Find  the  H.  C.  F.  of  x' -Zt?,  a?-  9a;,  a;' - 6x  +  9. 
x'-Z3?  =  i^(x-Z) 
ar'-9a;  =  a;(a;-h3)(a;-3) 
ic»-6x  +  9  =  («-3y. 
.•.H.C.F.  =  a;-3. 

Ex.  2.   Find  the  H.  C.  F.  of  6a:»y  -  12ay  4- 6y»  and  Sic'y* 

+  9x2/* -122/*. 

GxV  -  12x2/'  +  62/'  =  62/(x  -  yf 

3xy  +  9xy  -  12y*  =  Zy\3?  +  3x2/  -  ^)  =  Sj/'Car  +  4t/)  (x  -  y) 
r.K.C.¥.  =  Sy(x-y). 

Ex.3.   FindH.C.F.  of 

12a'b\  8aXa  -  b)\  16a'b\a  +  h)\  4a«  -  4a*b, 
H.C.F.  =  4a'. 

In  general, 

Separate  each  expression  into  its  prime  factors  ; 

Multiply  together  the  factors  common  to  all  the  expressions^  taking 
lach  common  factor  the  least  number  of  times  it  occurs  in  any  one 
expression. 


106  ALGEBRA. 

EXERCISE   37. 

Find  the  H.  C.  F.  of— 

1.  4a%  6ab\  (^a(a  +  6),  a'  -  b\ 

2.  5x%  15xY.  ^ix-yy,  x'-y\ 

3.  abc\  Sa'bc'.  IL  3^  —  3a:,  x"  -  9. 

4.  2Aa'x\  56aV.  12.  4a;=^  +  6x,  Qx'  +  9x. 

5.  Um\  42am'.  13.  a'  -  x\  a'  -  x\ 

6.  2ixy,  4Sax\  36a;.  14.  a;^  +  x,  a^  ^  1. 

7.  34aV,  Slaa:^.  15.  xy  —  y,  x^  —  x. 

8.  aV2/,  aV2/*z.  16.  4a' +  2a^  4a'  — a. 

17.  x'  +  x,  x''  —  l,  x^  —  x  —  2. 

18._a;'  +  a;  -  12,  a;^  -  a;  -  6,  a;'  -  6a:  +  9. 

19.  4a'a:  — 4aar',  8aV-8ax*,  4aV(a  — a;)l 

20.  2a;' -2a:,  3a:* -3a:,  ixi^x-iy. 
2h  Qx'  +  bxy  -  Ay\  ^x'  +  4:xy  -  Zy\ 

22.  3ar' -  5a:'*  -  2x,  4a:' -  5x' -  6a:,  a:' -4a;. 

23.  x'  -  81,  X*  +  8x'  -  9,  2a:*  +  17a:'  -  9. 

24.  b-d%  Sb-a'b-2a%  b'-a'b\ 

25.  1  -  a',  1  —  a^  3a  +  3a'  +  3a',  1  +  a'  +  a*. 

26.  xy-hx-y-1,  14a:' +  10a; - 24,  3(a:'-l)\ 


CASE   II. 

<f  Highest    Common    Factor  Determined    Indirectly    by 
Method  of  Long  Division,j 

119.  For  Polynomials  that  cannot  be  readily  factored 
the  H.  C.  F.  is  found  by  the  same  general  method  that  is 
used  in  arithmetic  to  determine  the  G.  C.  D.  of  large  num- 
bers. By  successive  divisions,  using  the  remainder  of  each 
division  for  the  next  divisor,  successive  pairs  of  smaller  and 


HIGHEST  COMMON  FACTOR  107 

smaller  numbers  are  found  which  have  the  same  H.  C.  F.  as 
the  original  numbers,  till  at  last  the  H.  C.  F.  is  determined. 
The  essential  parts  of  the  process  will  be  recalled  by  aid  of 
the  following  example : 

Find  the  G.  C.  D.  of  182,  299. 

182)299(1 
182 

117)182(1 
117 

65)117(1 
65 

62)65(1 
52 

13)52(4 
52 

.-.G.C.D.of  182,  299  is  13. 

120.  Principles  I.  and  II.  for  simplifying  the  Process 
of  finding  H.  C.  F.  by  the  Division  Method. 

Let  A  and  B  stand  for  any  two  algebraic  expressions. 
Then— 

I.  The  H.  C.  F.  of  A  and  B  is  the  same  as  the  H.  C.  F.  of 

A  and  mB,  or  A  and  —  >  provided  m  is  not  a  factor  of  A. 
m 

That  is,  one  of  two  algebraic  expressions  may  be  multi- 
plied or  divided  by  a  quantity  which  is  not  a  factor  of  the 
other  expression  without  changing  the  H.  C.  F.  of  the  ex- 
pressions. 

Ex.  The  H.  C.  F.  of  Sx,    Qax, 

is  the  same  as  H.  C.  F.  of       3a;,  12aa;, 
and  of  Sx,    6a;, 

the  H.  C.  F.  in  all  instances  being  3a?. 


108  ALGEBRA. 

II.  The  H.  C.  F.  of  a  pair  of  expressions,  A^  B^  is  the  same 
as  H.  C.  F.  of  the  pair  A,  B  —  rtiA. 

For  any  quantity  which  will  divide  both  A  and  B  will 
evidently  divide  B  —  mA. 

Conversely,  any  expression  which  will  divide  both  A,  and 
B  —  mA  will  also  divide  B. 

For  any  quantity  which  divides  A  will  divide  mA^  and, 
since  it  divides  B  —  mAj  must  also  divide  B. 

Hence,  the  H.  C.  F.  of  one  of  these  pairs  of  expressions  is 
the  H.  C.  F.  of  the  other  also. 

As  applied  in  the  method  of  finding  the  H.  C.  F.  by  the 
long  division  method,  this  principle  amounts  to  this,  that 
the  H,  C.  F.  of  the  divisor  and  dimdend  is  the  same  as  the 
H.  C.  F.  of  the  simpler  pair  of  quantities,  the  divisor^  and 
dividend  minus  quotient  X  divisor;  that  is,  of  the  divisor  and 
remainder. 

Principle  I.  enables  us  to  use  other  simplifications  in  the 
process  of  the  work. 

121.  Examples  Illustrating  the  Use  of  Principles  I. 
and  II. 

Ex.  1.  Find  H.  C.  F.  of  4x'  +  Sa;  -  10  and  4x'  -h  7a;'  -  3x 
-15. 

Divide  the  second  expression  by  the  first, 

4a:'  +  3x-10|4a;»  +  7a;^-   3a;  -  15  |  a;  +  1 
4ar'  +  3a;'  -  10a; 

4x'-h   7a;-15 

4a;' -h   3a;- 10 

4x-  5 

By  Principle  II.,  Art.  120,  the  H.  C.  F.  of  the  two  original 
expressions  is  the  same  as  that  of  the  simpler  pair,  4a;'  -f  3a; 
-10,  4a;-6. 

Proceeding  with  these, 


HIGHEST  COMMON  FACTOR.  109 

4a;-5|  4x'  +  3a; -  10  \x-\-2 


8x-10 
8x-10 

Since  4a;  —  5  divides  the  other  expression,  4a;'  -\-Zx  — 10, 
exactly,  it  is  the  H.  C.  F.  of  the  second  pair,  and  hence  of 
the  original  pair  of  expressions. 

.•.4a;-5  =  H.C.P. 

Ex.  2.  Find  H.  C.  F.  of  a;'  +  4a;'  +  5a;  4-  2  and  Sx*  +  15x» 
4-  12a;. 

The  second  of  these  expressions  is  divisible  by  3a;,  which  is  not  a  factor 
of  the  first  expression  ;  hence,  by  Principle  I.,  3a;  may  be  removed,  and  we 
proceed  to  find  the  H.  C.  F.  of 

a;'  +  4a;'  +  5a; .+  2  and  x"^  +  5a;  +  4. 

a;»  +  5a;  +  4  I  3:3  +  4a;'  +  5a;  +  2  I  a;  -  1 


a;'  +  5a;'  +  4a; 

-  a;'  +    a;  +  2 

-  a;'  -  5a;  -  4 

6a;  +  6 

We  have  now  to  find  the  H.  C.  F.  of  a;'  +  5a;  +  4  and  6a;  +  6.    But  by 
Prin.  I.,  Art.  120,  the  factor  6  may  be  dropped  from  6a;  +  6. 

a;  +  1  I  a;'  +  5a;  +  4  |  a;  -f  4 
a;'  4-    a; 
4a;  +  4 
4a;  +  4 


.  • .  a;  +  1  -  H.  C.  P. 

Ex.  3.  Find  H.  C.  F.  of  4ar»  -  4a;'  -  5a;  +  3  and  lOa:* 
-  19a;  +  6. 

To  render  the  first  expression  divisible  by  the  second,  by  Principle  I.  we 
may  multiply  the  first  expreasion  by  5,  which  is  not  a  factor  of  the  second 
«xpression, 


110 


0 

ALOEBBA. 

4a^- 

-    4a;'- 

5x  +    3 

5 

10a;'  -  19a;  +  6 

20a;3- 
20x3- 

-  20a;2  - 

-  38a;2  + 

25x  +  15 
12x 

2a: 

18x2- 

37x  +  15 

5 

90x2- 

185x  +  75 

9 

90x2- 

171x  +  54 

-7|- 

14x  +  21 

2x-    3 

J  10X2  - 
10X2- 

-  19x  -t-  6 
-15x 

5x-2 

-    4x  +  6 

-    4x-h  6 

.  • .  2x  -  3  -  H.  C.  F. 


122.  Arrangement  of  Work.  The  following  will  be  found 
a  more  compact  and  orderly  method  of  arranging  the  work  of 
finding  the  H.  C.  F.  of  two  expressions  (see  Ex.  3,  Art.  121) : 


lOx'-iGx  +  e 

Ax'- 
5 

-  4x^-     5x-f    3 

20a;^ 
20x' 

-20x^-    25a: +  15 
-38x^+    12a; 

18x^-   37a; +  15 
5 

l(V-15x 

90a;^  -  185a;  +  75 
90a;'^-171a;  +  54 

-   4x  +  6 

-71    -14a;  +  21 

-   4a; +  6 

HC.F.  =  2a;-   3 

2a; 


5a;-2 


123.  Removal  of  Simple  Factors.  It  is  important  for 
the  student  to  remember  that  if  either  one  or  both  of  the 
given  polynomials  whose  H.  C.  F.  is  sought  have  simple  fac- 
tors, these  simple  factors  are  to  be  removed  at  the  outset, 
and  their  H.  C.  F.  reserved  to  be  multiplied  into  the  H.  C.  F. 
of  the  remaining  polynomial  factors  as  found  by  the  division 
method, 


HIGHEST  COMMON  FACTOR.  Ill 

Ex.  Find  the  H.  C.  F.  of  6x*  -  30a:'  +  78x'  -  54a;  and   2x* 

-4a;*  +  8x'-6xl 

6x*  -  30a:'  +  78a;^  -  54x  =  6x(a;'  -  5a;'  +  13a:  -  9) 
2x'  -  42:'  +  8x'  -  6x'  =  2a:Xa:»  -  2x'  +  4a;  -  3). 
The  H.  C.  F.  of  6a:  and  2a;'  is  2a:. 

By  the  division  method  let  the  student  determine  the 
H.C.F.  of  a-'-5x^  +  13a:-9  and  ar*  -  2a:' -f  4a;  -  3. 

This  will  be  found  to  be  a:  —  1. 

Combining  these  results,  the  H.  C.  F.  of  the  two  original 
expressions  is 

2a:(a:-l). 

124.  The  General  Process  of  finding  the  H.  C.  F.  of  two 

expressions  by  the  division  method  may  now  be  stated  as 
follows : 

Arrange  the  given  expressions  according  to  the  descending  powers 
of  the  same  letters ; 

Remove  simple  factors  of  the  given  expressions^  reserving  their 
H.  C.  F.  as  a  factor  of  the  entire  H.  0.  F. ; 

Use  the  expression  of  lower  degree  for  divisor,  or,  if  both  are 
of  the  same  degree,  that  whose  first  term  has  the  smaller  coeffi- 
cient ; 

Continue  each  division  till  the  degree  of  the  remainder  is  lower 
than  the  degree  of  the  divisor ; 

Remove  from  each  remainder  each  factor  that  is  not  a  factor  of 
both  the  given  expressions; 

If  the  first  term  of  a  dividend  is  not  exactly  divisible  by  the  first 
term  of  the  divisor,  multiply  the  dividend  by  such  a  number  as  will 
make  the  term  divisible ; 

Continue  the  process  by  using  each  simplified  remainder  as  a 
new  divisor  and  the  last  divisor  as  a  new  dividend; 

The  first  divisor  to  divide  its  dividend  exactly  is  the  H.  C.  F.  of 
the  two  original  expressions. 


112  ALGEBRA, 

EXERCISE  38. 

Find  the  H.  C.  F.  of- 

X.  2x'  —  a:  -  3  and  4x'  -  Ax"  -  3a;  +  5. 

2.  6a;' -a: -12  and  6a;' -  13a;' - 6a;  +  18. 

3.  a;'  H-  a;'  +  a;  -  3,  ar'  -  3a;'  +  5a;  -  3. 

4.  3a;»  -  9a;'  +  9a;  -  3,  6a;'  -  6a;'  -  6a;  +  6. 

5.  6a;*  -  5a;'  +  6x'  +  5a;,  2x*  -  9ar'  -  9x'  -  2a;. 

6.  3x'  +  x'-a;  +  4,  3a;' +  7a;' +  a; - 4. 

7.  8a;'  +  2a;  -  3,  6a;»  +  5x'  -  2. 

8.  3a;' +  7a;' -5a; +  3,  2a^  +  3x^  -  7a;  +  6. 

9.  2x*  +  a;'  +  4x-3,  3a;*  +  2a;' - 2a;' +  3a; - 2. 

10.  x*-a;'-a;'  +  7a;-6,  a;*  +  a;' - 5x' +  13a; - 6. 

11.  a;*  +  3af^  +  9a;'  +  12a;  +  20,  x^  -f  6a;*  +  6a;'  +  8a;'  +  24a:L 

12.  2ar'  ■-  16a;  +  6,  5a;'  +  153;^  +  5a;  +  15. 

18.  2ar^  +  a;*  +  2af»  -  a;'  -  1,  5a;*  +  2af'  +  33;*  -  2x  +  1. 

14.  4ar^  -  10a;*  +  10a;'  -  lOx'  +  6a;,  4a;^  -  14x*  +  Sar*  +  lOx* 
-6a;. 

15.  3x*  +  2a;'y  +  2a;y  +  bxy"  -  2y\   6x*  +  ar'y  +  2xY  +  2xy» 

16.  3ar^  +  2a;*  -  8a;»  -Zt?  +  4x,  3ar^  -  10a;*  +  143;*  -  llx'  +  4x. 

17.  2a;*  -  3a;' +  2a;' -  3x  +  2,  3a;*  -  4a;' +  5x' -  6a; -f  2. 

The  H.  C.  F.  of  three  or  more  expressions  may  be  obtained 
by  finding  that  of  two  of  them ;  then  find  the  H.  C.  F.  of  this 
and  another  of  the  quantities ;  the  last  H.  C.  F.  thus  obtained 
is  the  one  required. 

18.  a;'-a;'-a;-2,  a;»  -  2a;' +  3x  -  6,  23;*  -  3x' -  a;  -  2. 

19.  2a;' -3a;' -5a; -12,  3a;*  -  73;*  -  2a;' -  12a;,  a;' -  9a;' +  27a; 
-27. 

20.  2x*  -  143;"  +  12a;,  2x*  +  6a;»  -  32x'  +  24x,  6a;*  -  3Qx* 
+  42a;'-18x. 


LOWEST  COMMON  MULTIPLE.  113 

LOWEST  COMMON  MULTIPLE. 

125.  A  Common  Multiple  of  two  or  more  algebraic  expres- 
sions is  an  expression  which  will  contain  each  of  them  with- 
out a  remainder. 

The  Lowest  Common  Multiple  of  two  or  more  algebraic 
expressions  is  the  expression'  of  lowest  degree  which  will  con- 
tain them  all  without  a  remainder. 

Ex.  1.  The  lowest  common  multiple,  or  L.  C.  M.,  of  3a', 
Qa\  iax'  is  12aV. 

Ex.  2.   The  L.  C.  M.  of  3x  and  4y  is  12xy. 

CASE  I. 
When  the  Lowest  Common  Multiple  may  be  found 
directly  by  Inspection. 

126.  L.  C.  M.  of  Monomials. 
Take  the  L.  C.  M.  of  the  coeffiQients  ; 

Annex  each  literal  factor  that  occurs  in  any  of  the  given  expressions  j 
giving  the  letter  the  highest  exponent  which  it  has  in  any  one  expression. 

Ex.  Find  L.  C.  M.  of  4aV,  5a3^,  lOa^x^y. 
The  L.  C.  M.  of  4,  5, 10  is  20. 
The  highest  exponent  of  a  is  2. 

"  "  "  X  is  5. 

"  "  "  y  is  1. 

.•.20aV2/  =  L.C.M. 

127.  L.  C.  M.  of  Polynomials  readily  Factored. 
Ex.  Find  L.  C.  M.  otx^  —  Sx\  x^  —  9x,  x^  —  6x-bd. 

x*-^x'  =  7^ix-^) 

x'-9x  =x(x-^^)(x-Z) 
x'-6x  +  d  =  (x-Sy 
.'.L.C.U.  =  xXx  +  Z)(x-Z')\ 

Hence,  in  general, 


114  ALGEBRA. 

Separate  each  expression  into  its  prime  factors  ; 
Take  the  product  of  all  the  different  factors^  iising  each  factor  the 
greatest  number  of  times  which  it  occurs  in  any  one  expression. 

EXERCISE   39. 

Find  the  L.  C.  M.  of— 

1.  3a^6,  2ab\  8.  12a%  lQab\  2Aa'b\ 

2.  6x'y,  Sy'z.  9.  7a\  2ab,  Gb\  21. 

3.  12aV,  2dY.  10.  3x^  8,  Qx%  \2xy\ 

4.  16xy,  12xy.  11.  2x{x  +  \),  x^-l. 
6.  2ac,  36c,  4a&.  12.  a"  +  ab,  ab  +  b\ 

6.  Za%  Aac\  6b'c.  13.  7x\  2x'-6x. 

7.  42a;y,  2Sy'z\  14.  ar'-l,  af'-l. 

15.  x'  -  y\  ^  -  Sxy  +  2y\ 

16.  32:^- 3a;,  6a;' -122; +  6. 

17.  ax^Cx  —  yy,  bxy(x^  —  y^). 

18.  a;'  -  3a;  -  40,  a;'  -  9a;  +  8. 

19.  3a;'^  +  2a;  -  8,  ^^  +  x- 12. 

20.  a'  -  6^  a'  -  b\  a'  +  b\ 

21.  6a;'  +  6a;,  2a;'  -  2a;',  Sx"  -  3. 

22.  a'b  4-  ab',  a'b  -  ab\  3a'  -  W, 

23.  2a;'  +  a;-l,  4a;' -1,  2a;' +  3a;  +  1. 

24.  32;^  -  3,  6a;'  -  12'a;  +  6,  2a;'  +  2a;'  +  2a;. 

25.  12x' -  2x' -  140a;,  18a;' +  6a;  -  180,  6a;' -  39a;' +  63a;. 

26.  l-a;  +  a;'-a;',  1  +  a;  +  a;' +  a;',  2a;-2a;'. 

27.  (a;-l)',  7a;2/'(a;' -  1)',  Ux^yix  +  lf. 

28.  18a;'-12a;'  +  2x,  273;^- 3a;',  18x' -  24a;' +  6a;. 

29.  36x*-81a;',  16a;^  -  48a;*  +  36a;',  24a;*  +  72a;' +  543;^. 

30.  a;'  -  1  -  2a  -  a',  a;'  -  1  +  a'  +  2aa;,  a;'  +  1  -  a'  -  2a;. 

31.  (a;-l)(a;  +  3)',  (a;  +  1)' (a;  -  3),  (a;'-l)',  a;' -9. 


LOWEST  COMMON  MULTIPLE.  115 

CASE    II. 

Lowest  Common  Multiple  Determined  Indirectly  by  the 
Division  Method. 

128.  If  it  be  required  to  find  the  L.  C.  M.  of  expressions 
which  cannot  be  factored  readily,  we  proceed  in  general  as  in 
arithmetic  when  finding  the  L.  C.  M.  of  two  large  numbers ; 
that  is,  we  first  find  the  H.  C.  F.  of  the  two  numbers. 

Thus,  to  find  the  L.  C.  M.  of  182,  299  by  the  division 
method,  the  G.  C.  D.  is  found  to  be  13. 

Then,  since     182  =  13X14,        299  =  13X23, 

13  1 13X14  ,  13X23 
14  ,  23 

.  • .  L.  C.  M.  =  13  X  14  X  23  =  182  X  23. 

In  brief,  we  find  the  G.  d  D.  of  the  two  numbers,  divide 
one  of  the  numbers  by  this  G.  C.  D.,  and  multiply  the  quo- 
tient by  the  other  number. 

Similarly,  to  find  the  L.  C.  M.  of  two  algebraic  expressions 
which  cannot  be  readily  factored,  we  first  find  the  H.  C.  F.  of 
the  two  expressions  by  the  division  method. 

Thus,  to  find  the  L.  C.  M.  of  4a;'  +  3a;  -  10  and  4ar'  +  loc" 
—  3a;  —  15,  we  first  find  the  H.  C.  F.  by  the  division  method ; 
this  is  4a;  —  5. 

Then  4a;'  +  3a;  - 10  =  (4a;  -  5)  (a;  +  2) 

4ar*  +  7x' -  3a;  - 1 5  =  (4a;  -  5)  (a;' 4- 3x  +  3) 

.  • .  L.  C.  M.  =  (4a;  -  5)  (a;  +  2)  (a;'  +  3a;  +  3) 

=  (4a;'  +  3a;-  10)  (a;"  +  3a;  +  3). 

Hence,  in  general, 

Find  the  H.  C.  F.  of  the  two  given  expressions ; 
Divide  one  of  the  expressions  by  the  H.  C.  F.y  and  multiply  the 
quotient  by  the  other. 


116  ALGEBRA. 

EXERCISE  40. 

Find  the  L.  C.  M.  of— 

1.  x'  -  5a:  —  2  and  x'~x-\-Q. 

2.  Sx^-hx^-x^  4,  S7^^7x'  +  x- 4. 

3.  6x'  -  Sx'  -  9a:  -  3,  6a;*  +  9a:'  +  9a:'  +  3a:. 

4.  6a:'  -  3a:'  -  10a:  +  5,  8a:'  -  4a:'  +  20a:  - 10. 

5.  8x*-20a:'-14a:'  +  5x  +  3,  4a:'-3a:-l. 

6.  12a:' -8a:' -27a: +18,  18a:' -  27x' -  8a:  +  12. 

7.  3a:'-2a:'-l,  4a:'-5a:  +  l. 

8.  2a:'  +  3a:'  -  a:  +  2,  3a:'  -  x'  -  9a:  +  10. 

The  L.  C.  M.  of  three  or  more  expressions  may  be  obtained 
by  finding  that  of  two  of  them ;  then  finding  the  L.  C.  M.  of 
this  result  and  the  third,  expression.  The  last  L.  C,  M,  thus 
obtained  is  the  one  required. 

9.  a:'  -  7a:  +  6,  a:'  4-  7a:'  -  36,  a:'  -  31a:  +  30. 

10.  4a:'-13x  +  6,  4a:' - 4a:' - 5a:  +  3,  3a:'  +  7aJ'-4. 

Find  the  H.  C.  F.  and  L.  C.  M.  of— 

11.  20a'6'c',  35a'6'(i',  14a'6'c',  lOa'6'c'cZ'. 

12.  3a'5(a:'  -  1)',  6a6'(5a:'  +  3a:  -  2)',  9(3i^  4-  5a:  +  2)». 

13.  a:* -f  2a:' +  a:' -  4,  a:*  -  a:' +  4a:  -  4. 

14.  4a:'  -  3a:  -  1,  2a:'  -  3a:'  +  1,  Qx'  -  .t'  -f  1. 

15.  3a:' +  2a;' -7a: +  2,  4a:'-12a:  +  8,  *J^ -M2-.f:*  -  llaJ  4- 2. 


CHAPTER    X. 
FRACTIONS. 

129.  Origin  and  Use  of  Fractions.  It  is  sometimes  neces- 
sary to  indicate  the  division  of  one  algebraic  expression  by 
another,  but  apart  from  this  it  is  often  useful  to  do  so.  For 
when  a  number  of  indicated  quotients  is  combined  in  a  pro- 
cess, cancellations  and  other  simplifications  are  possible  before 
making  the  final  reduction,  and  in  this  way  much  labor  is 
saved. 

130.  A  Fraction  is  the  quotient  of  two  algebraic  expres- 
sions indicated  in  the  form  -• 

.  b 

This  form  of  indicating  a  quotient  is  preferred,  since  it 
enables  us  readily  to  discriminate  the  parts  of  a  fraction 
from  the  rest  of  an  expression,  and  hence  to  compare  the 
parts  of  difierent  fractions  to  the  best  advantage. 

Thus,  the  fractional  part  of  the  expression  is  more  readily 

x  —  1 
perceived  in  x'  H 1-  5,  than  in  cc^  +  (x  —  1)  -^  (a;  4-  2)  +  5. 

x  +  2 

131.  The  Numerator  is  the  dividend  part  of  the  indicated 

quotient,  or  part  above  the  line;  the  divisor,  or  part  below 

the  line,  is  called  the  Denominator.     The  numerator  and 

denominator  are  called  the  Terrris  of  the  fraction. 

5x  -\-  2 
If  (5a;  +  2)  -j-  Sx^  be  written  as  a  fraction,  we  have  — — — ; 

that  is,  the  dividing  line  of  a  fraction  takes  the  place  of  a 
parenthesis,  and  hence  is  in  effect  a  vinculum. 

132.  An  Integral  Expression  is  one  which  does  not  con- 
tain a  fraction;  as,  3x^  —  2y, 

117 


118  ALGEBRA. 

133.  A  Mixed  Expression  is  one  which  is  part  integral, 
part  fractional. 

134.  Sign  of  a  Fraction.  A  fraction  has  its  own  sign, 
which  is  distinct  from  the  sign  of  both  numerator  and  de- 
nominator. It  is  written  to  the  left  of  the  dividing  line  of 
the  fraction. 

GENERAL  PRINCIPLES. 

135.  A.  If  the  num,erator  and  denominator  of  a  fraction  be 
both  midtiplied  or  both  divided  by  the  same  quantii^j  the  value  of 
the  fraction  is  not  changed. 

This  principle  is  seen  to  be  true  at  once,  since  the  terms  of 
a  fraction  are  a  dividend  and  a  divisor.  It  is  a  useful  exer- 
cise, however,  to  derive  it  from  the  fundamental  laws  of 
algebra  (see  Art.  33). 

CL 

-.  =  a-^b  =  a^bXm-^m 

0 

=  aXm^b-^m    (Comm.  Law.) 
aXm 


b 

■     lib 

aXm 
bXm 

I- 

m  = 

am 
b 

a 
b'^ 

m  = 

a 

bm 

Similarly, 
and 


136.     B.  Law  of  Signs.     By  the  laws  of  signs  for  multi- 
plication and  division  (see  Arts.  52,  64), 

a      —  a        a      —  a a      a  a       a 


6—6        6         6        —b     be  —bXc      —bX—c 


FRACTIONS.  119 

Or,  in  general, 

The  signs  of  any  even  numjjer  of  factors  of  the  numerator  and 
denominator  of  a  fraction  may  he  changed  without  changing  the 
sign  of  the  fraction. 

But  if  the  signs  of  an  odd  number  of  factors  be  changed^  the 
sign  of  the  fraction  must  be  changed. 

TRANSFORMATIONS  OP  FRACTIONS. 

I.  To  Reduce  a  Fraction  to  its  Lowest  Terms. 

137.  A  fraction  is  in  its  lowest  terms  when  its  numerator 
and  denominator  have  no  common  factor. 

138.  Direct  Reduction.  When  the  terms  of  the  fraction 
are  monomials,  or  polynomials  readily  factored, 

Resolve  the  numerator  and  denominator  into  their  prime  factors, 
and  cancel  the  factors  common  to  both. 

Ex.  1.   Reduce — -  to  its  lowest  terms. 

4Sa'xy 

Divide  both  numerator  and  denominator  by  12aV  (see 
Art.  135). 

36aV         3a 


Ex.2. 


*  48aV2/"      Axy" 
9ab-12b'       36(3a-46)_36 


12a'-16a6      4a(3a-46)      4a 

The  student  should  notice  particularly  that  in  reducing  a 
fraction  to  its  lowest  terms  it  is  allowable  to  cancel  a  factor 
which  is  common  to  both  denominator  and  numerator,  but 
that  it  is  not  allowable  to  cancel  a  term  which  is  common 
unless  this  term  be  a  factor. 

Thus,  —  reduces  to  -5 

ac  c 


120  ALGEBRA. 

a  ~\~  X 
but  in  ?  a  of  the  numerator  will  not  cancel  a  of  the 

a +  2/ 

denominator. 

This  is  a  principle  very  frequently  violated  by  beginners. 

139.  Finding  H.  C.  F.  of  Numerator  and  Denominator 
by  Division  Method.  When  the  numerator  and  denomi- 
nator of  a  fraction  cannot  be  factored  by  inspection, 

Find  the  H.  C.  F.  of  the  numerator  and  denominator  by  the 
method  of  Art.  12 Jf.^  and  divide  both  numerator  and  denominator 
by  their  H.  C.  F. 

Ex.  Simplify  — ; 

^    ^   9a;'-22a:-8 

The  H.  C.  F.  of  the  numerator  and  denominator  i&  ^und 
to  be  ^x'-Ax-2. 

Dividing  both  numerator  and  denominator  by  this> 

6x^-110:^  +  2      2a:- 1     ^     ,^ 

— — = )  Result. 

9ar'-22a:-8       3a;  +  4 

EXERCISE  41. 

Reduce  to  their  simplest  form — 

1  ^^.  5        2a  8(^--l). 
'  12aV'                     *  4a' -2a'                   *   12a: -12* 

2  1?^.  6       ^^"^^  10    i^-Zll)!. 
*  153:^3/^                     '   6aa:-12a?/  *    \^(x-yy 

3a'a;  '         Ax-\'Ay  a?h -{- ah" 

6a'-9a'x*  *   Aax  +  Aay'  *   2a'^6-2a6''' 


12. 


72a:VV  JiZ"J(!_. 

96a:2/V*  '    (a; +  2/)'*                "    ^xSj-12xi^ 

L3    ^^  ~  ^^y  14    49a:'  -  QAf 

is^-dxy"'  .'  14a:'-16a:'y' 


FJRACTIONS.  121 


^^  (^-yy(^  +  yy  23.     ^*~^^ 


16. ^ .  24.  — ^ 

4x»  -  2a;y  -  1 2y'  a;*  +  xy  +  y* 

^^     Qx'-xy-2y\  ^     a:'-2x-l 


67^-7xy  +  2f  '    ar'-2x*  +  l 

(a  +  6)'-c'  18>  +  19a;V-12v* 

*   a'-(6+c)^* 

,^     l-(a-a;)' 

19.  ^^ —'  27. 

„^    4-(a  +  6)» 

20. ^- — er-  28. 


27x*  +  6xy  -  Sy' 

a^-Sx'  +  17a:  - 10 

a;*-2x'-4u;'^  +  lla;-6 

2ay^  +  (ix"  -  9a 

St^-Sx-IS 

3.r3  -f  4.7-2  _  a;  +  6 

2a:3  _^  7^2  _^  4^.  _  4- 

2x^-llx'-9 

(a-2)'-6» 

2j    m«-2ax-24a' 
4a;'-2ax-6a' 

22  ^-^  30 . 

'  ic'y'  +  2^2/'  +  43/'  *  4x*  +  llx*  +  81 

a;*  —  z'  —  4  —  2x1/  —  4z  +  y* 

ol» • 

z^  —  x'  —  4  —  2yz  —iz  +  y' 


n.  To  Reduce  an  Improper  Fraction  to  an  Integral 
OR  Mixed  Quantity. 

140.  An  Improper  Fraction  is  one  in  which  the  degree  of 
the  numerator  equals  or  exceeds  the  degree  of  the  denomi- 
nator. 

Since  a  fraction  is  an  indicated  division,  to  reduce  an 
improper  fraction  to  an  integral  or  mixed  expression, 

Divide  the  numerator  by  the  denominator  ; 
If  there  he  a  remainder^  write  it  over  the  denominator ^  and  annex 
the  remit  to  the  quotient  with  the  proper  sign. 


122  ALGEBRA. 

Ex.  1.   Reduce  —  to  an  integral  or  mixed  expres8ioi\ 


•-  x^y  —  xy^ 


.•.^-^»      xy 
x^y 

+  2 

X.2. 

Reduce 

7?  +  X  +  2 

a:»  +  4a:' 

5 

rB»  +    a;'  +  2a; 

xy^  -y^ 
xy"^  4-  .v' 


x  +  y' 


x*  +  x  +  2 


BesvlU 


x  +  3 


3x^  -  2a:  -  5 
3x-'  +  3a;  +  6 


-  5a;  -  11 

;.^  a;»  +  4a:'-5^^^3_     5a:  4- 11    ,  j^^^ 

a:'  +  a;  +  2  a;2  +  a:  +  2 

When  the  remainder  is  made  the  numerator  of  a  fraction  with  the  minus 
sign  before.it,  as  in  this  example,  the  signs  of  terms  of  the  remainder  must 
be  changed,  since  the  vinculum  is  in  effect  a  parenthesis  (see  Art.  48). 

EXERCISE   42. 

Reduce  each  to  a  mixed  quantity — 

^    x'  +  Bxy-2f-l 

6. 

7. 


8. 


1. 

a;^-2x  +  3 

X 

2. 

4ar'  +  6a:  -  5 

2x 

o 

lOftV  -h5ax-7  —  a 

bax 

4. 

7?-Zx'  +  x-\ 

x-\-y 

Zx' 

-13a; -28 

a:^-3 

7f- 

ar"  -  a;  -h  2  - 

-a 

a;-l 

x' 

+  1 

x'-x-l 

2x*  +  7 

x'  +  x-^l 

x*-{-x''-x- 

-1 

7^  +  2 

9a» 

3a' -26 

T^-hx'-4x  +  7 

FBACTIONS. 

14. 

2a* 

a  +  6 

15. 

a^^-x^  +  x*- 

-2x 

x'^1 

16.*  — i— . 

l+x 

17. 

1 
l  +  a;-x» 

18. 

\  8 

123 

9. 
10. 
11. 
12. 

■iq 

x  +  3  '"   2  +  x-x' 

ni.  To  Reduce  a  Mixed  Expression  to  a  Fraction. 

141.  It  is  necessary  simply  to  reverse  the  process  of  Art. 
140  in  order  to  reduce  a  mixed  expression  to  a  fraction. 
Hence, 

Multiply  the  integral  expression  by  the  denominator  of  the  frac- 
tion, and  add  the  numerator  to  the  result,  changing  the  signs  of  the 
terms  of  the  nuw^erator  if  the  fraction  he  preceded  by  the  minus 
sign; 

Write  the  denominator  under  the  result. 

fi 2 

Ex.  1.   Reduce  a  —  1  H to  the  fractional  form. 

a-rS 

q-  1  +  gLl-2  ^  g'  +  2a  -  3  +  g  -  2  ^  g^  +  3a  -  5   j^^^^ 
g+3  g+3  a+3 

Ex.  2.  a:  +  y  -  ^-±Jl. 
x-y 
^  (a;  +  y)  {x  -  y)  -  (x^  +  y^) 

x-y 
^  aJ  _  y2  _  a.2  _  yi  ^  -  2.v' 
x-y  x-y 


2.V' 
y  -  X 


,  Result 


*  To  three  integral  terms. 


124  • 

ALOEBBA. 

EXERCISE  43. 

Reduce  to  a  fraction — 

1.  a-H--. 
a 

8.  ^-^'  +  a      1. 

2a 

2.  x  +  l-H— i— . 
x  —  1 

9.  — ^  +  a  +  2. 
a  — 1 

3.  ^  +  x-l ^ 

ic  — 1 

■   -t-i— ■ 

4.4^-2    y-^. 

2i+l 

11.  a;  -  a  —  ^^~^  +  y, 

x  +  a 

5.  a     6+     2*;. 
a +  26 

12.1      (.      .'+^;j. 

6.-     1         '~^     • 

7?  +  X-^l 

''■''-H'-.li)] 

7.  a     a:  +  l      ''~-^. 
a-\-x 

^^-'-f-^^/J- 

15.^-{-.'-[x  +  l-^]} 


IV.  To  Reduce  Fractions  to  Equivalent  Fractions 
OF  THE  Lowest  Common  Denominator. 

142.  Since  by  Art.  135  we  may  multiply  the  numerator 
and  denominator  of  a  fr^ion  by  the  same  quantity  without 
altering  the  value  of  the  fraction,  we  can  use  the  same  pro- 
cess as  in  arithmetic  for  reducing  fractions  to  their  lowest 
common  denominator. 

It  is  supposed  at  the  outset  that  each  fraction  has  been 
reduced  to  its  lowest  terms. 


FRACTIONS.  126 

(  Find  the  lowest  common  multiple  of  the  denominators  of  the  given 
fractions  ; 

Divide  this  common  multiple  by  the  denominator  of  each  fraction ; 
Multiply  each  quotient  by  the  corresponding  numerator;   the 
^,  results  will  form  the  new  numerators ; 
\\jVrite  the  lowest  common  denominator  under  each  new  numerator. 


m-i- 


2        3         5 

Reduce  - —  >  — —  j  - — -  to  equivalent  fractions  hav- 
Sax    4a'x     Qaa^        ^ 

ing  the  lowest  common  denominator. 

The  L.  C.  D.  is  12a^x\ 

Dividing  this  by  each  of  the  denominators,  the  quotients  are  4ax,  3a:,  2a. 
Multiplying  each  of  these  quotients  by  the  corresponding  numerator  and 
setting  the  results  over  the  common  denominator,  we  obtain 
Sax  9x  IQq 

Ua^'x^ '    12a2a;2 '     Ua'x' ' 

Ex.  2.  Reduce  to  their  lowest  common  denominator > 

x  —  y 
X  1 


x  +  y     x^—y' 

The  L.  C.  D.  is  x^  -  y\ 

Dividing  this  by  each  denominator,  the  quotients  9.re  x  +  y,  X  —  y,  1. 

Multiplying  each  quotient  by  the  corresponding  numerator  and  setting 

ion  denominator,  we  ol 

xy       x^  —  xy      1^ 


7?.-y^'     x'-y^'    x'-y^ 

EXERCISE  44. 

Reduce  to  equivalent  fractions  having  the  lowest  common 
denominator — 


1.  — ,  — •  4. 


2x    , 
9' 

bx 
6' 

12a 
56' 

7     a 
lO'  b 

1 
2a6» 

2 
■'  a'6  ' 

1 

2ac    46c    3a6 


„....«,  ^     2     ^   13 


2       3^1 
—  >  — »  2o,  -• 

3a'    4ax  x 


126 

(5^c     ab    be    ad 

bd    cd    ad    be 

«    1      ^ 

o.   ) 


ALGEBRA. 


1        2        ^ 


a'  —  a         a  —  1 

..        a:        ,    1        1 
10. )  1,  -> 


11. 


l-\-  X  X     X-^7? 

X  1 


x'-l    x'-l 

N12.  ^,  ^,  ^. 
VL^2  — 9    2a;  +  3    > 

13.  m,  


16. 
17. 

18. 
19. 


a^b  +  a6'     a^b  -  a6' 

1         5^ 

3x  -  6  '  2x  4- 


4'x^ 


21. 

^^ 
.23. 


(C  +  l 


^'  12, 

x^-^x-Q  x^-^Ax  + 


x-7?    3  +  3a;  2-2a; 

on       1  2  3 

20.  > » 

4-a;'    2a;  +  x'  4-2a; 

a:-l_ 


-'  «^ 


,  4, 


a6 


111 


^--^2a;'4-3a;-2     a:'  +  3a;+2     2a:'  +  a;-l 

PROCESSES  WITH  FRACTIONS. 
I.  Addition  and  Subtraction  op  Fractions. 

143.  By  the  Distributive  Law  (Art.  33),  inverting  the  order 
of  the  expressions, 

a      b  _  a  +  b 
c       c  c 

Hence,  to  add  or  subtract  fractions, 

Reduce  the  fractions  to  their  lowest  common  denominator; 


FRACTIONS. 

Add  their  numerators^  changing  the  signs  of  the 
any  fraction  preceded  by  the  minus  sign  ; 
Set  the  sum  over  the  common  denominator; 
Reduce  the  sum  to  its  lowest  terms. 


Ex. 

1.  i. 

X 

x  +  1       x  +  2 

-. 

_(x+l)(x^2)- 

2x(x  +  2)+x(x+l^ 

x{x  +  l)(x  +  2) 

x'+Sx  +  2~2x' 

-Ax  +  x'  +  x 

x(x+l){x-{-2) 

2 

x(x  +  l)(x-t2-) 

Ex. 

2.   - 
a 

a             ,1,1 

—  1               a^  —  a       a 

a          a          1 

a-1       1      a'~. 

a      a 

o'  —  a'  +  a'  +  1  + 
a(a-l) 

a-1 

—  a*  4-  2a'  +  a 
a(a~l) 

-a'  +  2a  +  l 
a-1        ' 

Result. 

Collect— 

EXERCISE  45. 

1. 

t 

X      Sx' 

Ax 

_2 

x 

2. 

2 
3a 

Aax      X 

^    Za-b 
■    '•      2a 

a-ib 
36 

3. 

5 

2ac 

2         1 

Sab      be 

a  +  26 
^-     2ab 

6a-l 
6a' 

128  ALGEBRA. 

^    2a'x  +  3  Sa  +  x  ^^       1 


4ax*  6a;  a  —  b       a  -f  6 

^    ,    ,   4x-3       3x  +  2            ,,     ic-1       x  +  l 
8.  IH —  •  11. 

6  5  a;  +  l       x-1 

8  6  X  3x' 

3a  —  46      2a  —  6  —  c       15a  —  4c 
^^•""2  3"""^        12       • 

3x-l_x-6^+2  2x-4, 

7  4  28  12 

2x^-32       xz'-y'z      y-Sxz"       2 
*       Sx'y  2xy'  Qxh         3* 

..a             6                      o.       3x           2x      ,     10a; 
16. -•  21.  — —- -  + 


a-b       a  +  b  x  +  2       x-2      x'-A 

17.^ ^.  22.-^-2'      2^' 


X  —  3       x  —  4  x'  +  x  x'  —X 

'  x-2      x  +  2  '  &C-S      Ix-Vl     ei'-e 


(771-1/       m'-l  9-a'      3+a       3-a 

2.3  7a; 


X        .  ^  X 


20.  -  +  1 -•  25 


ic-l  x+l  2x-l      4x  +  2      4a;*''-l 

26.-^  +  2-^^-^^. 
a;'-l  a;  +  l       a;-l 

27.-^—^.4-      ^ 


a;  +  l       x  +  2       a;  +  3 
oo         a;  +  2  a;-3     ,         2x4-5 

28.    _   „       ■        —  — ;        T  "I 


29. 


2x'  +  x-l       4x^^-1       2x^  +  3x4-1 

b  ab  ab^ 

a  +  b  ~  (aT"67  ~  '(a~+W 


FRACTIONS.  129 


30. 


2x^-x-l       2x'-hx-3      4x^-h8a;  +  3 


a;"'*  —  2/'      2a;      23/  2xy 

32.   ^^~3/  _^  _li^J/_  . 
'  x-\-2y       x^  —Ay^       ^  ~  % 


g  — 6  ,  o  _  g  +  6 

a^  —  h^      a  +  h  a  —  b 

_        2  a; -3  a;' 

35. — 


a; +  4      a;' -4x4-16      a;' +  64 

2  2  1 


a;'  — 3x  +  2        a;'  — a;-2      x'  — 1 
6x  7  '     26 


2(x-3)'       3x4-9      4x^-36 

38  i-\^^-r-J—i ^1_ll_^L.. 

'  X      1x4-1      Lx'-x4-l       x4-lj        i      ar'4-1 

l_|x»-6x-3      r_l ^_-]) 

2      1      2a;^-2  Lx-1      x'  +  x4-lJ) 

x-2      Lx       x^-3x  +  2        U-l        /J 

144.  Changing  Signs  of  Factors.  The  process  of  reducing 
fractions  to  their  lowest  comm  n  denominator  is  frequently 
simplified  by  changing  the  sign  of  one  or  more  of  the  factors 
of  a  denominator,  at  the  same  time  making  the  necessary 
change  in  the  sign  of  the  fraction.  It  is  to  be  remembered 
from  Art.  136  that  if  the  sign  of  an  even  number  of  factors 
be  changed,  the  sign  of  the  fraction  is  unchanged ;  but  if  the 
sign  of  an  odd  number  of  factors  be  changed,  the  sign  of  the 
fraction  is  changed. 


130  ALGEBRA. 

Ex.  1.  Simplify  - — -  + 


a; -hi       1  —  x 

The  factors  oi  x^  —  1  are  x  -\-  \,  x  —  l.  Hence,  if  the  sign  of  the  de- 
nominator, \  —  X,  be  changed,  it  will  become  x  —  \,  and  be  a  factor  of 
x^  —  1,  But  by  Art.  136,  if  the  sign  of  1  -  x  be  changed,  the  sign  of  the 
fraction  in  which  it  occurs  must  also  be  changed.     Hence,  we  have 

x"^  X  X     _  x^  +  x"^  -  x  ^  x^  +  X  ^     Zx^    ^  ^^^ 

a:'*  -  1      a;  +  1      a;  -  1  x^  -  1  x'  -  1 

Where  the  differences  of  three  letters  occur  as  factors  in  the  various 
denominators,  it  is  useful  to  have  some  standard  order  for  the  letters  in 
the  factors.  It  is  customary  to  reduce  the  factors  so  that  the  alphabetical 
order  of  the  letters  be  preserved  in  each  factor,  except  that  the  last  letter 
be  followed  by  the  first. 

This  is  called  the  cyclic  order. 

Thus,  a  —  bf  b  —  Cf  c  —  a  obey  the  cyclic  order. 
Ex.  2.  Simplify 


(a-6)(c-a)       (a-b)(c-b)      (c-b)(a-c) 

Changing  c  -  b  to  b  —  c,  and  a  —  c  to  c  -  a  where  they  occur,  we 
obtain 

1 I +  1 

(a  -b)  (c  -a)      (a  -  6)  (6  -  c)       (6  -  c)  (c  -  a) 

^ b  —  c  —  c  +  a  +  a  —  b 

(a  -  b){b  -  c)  (c  -  a) 

2«-2«  -2  -Sum. 


{a  -  6)  (6  -  c)  (c  -  a)      (a  -  6)  (6  -  c) 


EXERCISE   46. 


Collect- 


3^     +^+      1 


x'-l      1-x       1  +  x 
2.      2a^+_L,+     2 


a^  —  b^       a  +  b      b  —  a 


FRACTIONS.  131 


'  x^  —  4y^       2y -\- X       '^y  —  x 


x-l      1  +  a;       1— ic* 
6        5  3a  4 -13a 

'  l  +  2a       l-2a       4a^-l' 

'    ^  —  y"       x-\-y       y  —  x 

8-8a       4a  +  4.      8a'-8^ 
9j  +  ^_+     5.  1  3 


a;       x  —  1       1—x^       x  -{-1      x  +  x* 
10.  1  1  1 


(x-2)(3-a:)       10-7a;4-a;'^       (5-x)(aj--3) 

2                               3  4 

11.  z ;t-7; ^  -  z ::rzz tt  + 


(a  -  3)  (6  -  2)        (a  -  2)  (2  -  6)        (a  -  2)  (3  -  a) 

+ ^ 

(a  -  3)  (2  -  6) 

5a  5a  a 

l-^*   T:^  ITT    I 


13. 


6a-18       27-3a'      4a  + 12 

2b  +  a  _  26  — g  _  46a;  —  2a' 

x  +  a         a  —  x  x^  —  a^ 


,,      x  +  1  2x~l     ,        2 

14. —  — -—  + 


6x-6       12a; +  12       3  -  3x'      12a; 

X        ,        X           .       1  — a;    ,     15x  +  3 
15.  —  +  — 1 r 


16.* 


2a:-6        3a;  +  9  6x         18a;-2ar» 

7?-x-^        a;'  +  4x  +  3         15a; 
a:'  +  5a;  +  6       a;»-4a;  +  3       O-a;** 


*  Eeduce  before  adding. 


132 


ALGEBRA. 
4a' -6' 


+ 


a'-2a5  4-36« 


.      a'-b'  2a' -Sab -2b'      a' 

1-x'  ,      x'-O         x'-4x  +  3 


3a6  +  26* 
2x 


/T^f'^      3(a;  +  3) 
19.       ^^  +  ^       ■ 


5(x-3)'^        5x^-45 
X  4  — X 


5a; +  6 


20. 


8a;-x'-15 
b 


Ix 


10 


(a  —  6)  (a  ~  c) 


+ 


(6-c)(6-a) 
6' 


+ 


(c  —  a)  (c  —  6) 


22. 


/ 


{(x-b^{a-c)       (6-c)(6-a)       (c-a)(c-6) 

y  4-z  z  +  a;  x  +  i/ 

(2;-2/)(a;  — z)      {y  —  z){y  —  x)      (z  —  x)  (z  —  y) 

yz ,  zx  _^  xy 

(x-y)(x  —  z)      (y-z)(y-x)      (z-x)(z-y) 

1+Z  .  1+m  ,  1+n 


'^ 


+ 


+ 


m)  (^  —  n)       (m  —  n)  (m  —  Z)      (n  —  Q  (n  —  m) 
n.  Multiplication  of  Fractions. 


(I  c 

145.  To  find  the  product  of  any  two  fractions,  -  and  -  ^  we 

b  a 

may  proceed  thus : 

^X-^  =  a^bXc-^d  =  aXc-^b^d    (Art.  33) 
b      a 


_aXc  ,    , 


aXc 


b  bXd 

Hence,  to  multiply  fractions. 

Multiply  the  numerators  together  for  a  new  numerator^  and  mvl- 
tiply  the  denominators  together  for  a  new  denominator,  canceling 
factors  that  are  common  to  the  two  products. 

This  reduces  the  multiplication  of  fractions  to  the  multi- 


FRACTIONS.  133 

plication  of  integral  expressions,  and  enables  us  to  use  again 
3ur  knowledge  of  the  latter  process. 

'   M      126V       6a^2/ 

^  2  X  10  X  4a»6Vy»  _  2y*_ 
6  X 12  X  6a*6V2/  ~  9a  ' 

Ex.2.  ^±^X-^-^X      ^"' 


cc*  +  xy'^       {x  +  J/)' 
=  ^  +  ^  X  (a;  +  2/)(a^-2/)  w 4^ 


in.  Division  of  Fractions. 

146.  To  divide  any  fraction,  -  >  by  any  other  fraction,  -  >  we 

0  d 

may  proceed  thus : 

Let  a;=-)    y  =  -  ,* .  bx  =  a,      dy  =  c 

b  d 

X      ad      a  ^ ,  d    ,    .    X      a      c 
y      be       0      c  y      b      d 

Hence,  to  divide  one  fraction  by  another, 
Invert  the  divisor  and  proceed  as  in  multiplieation. 

This  reduces  division  of  fractions  to  the  already-learned 
process  of  multiplication  of  fractions. 


=  c 

bx  _  a 
'    '  dy      0 

a 
b'^ 

c  _a      d 
'd~b^c 

134  ^  ALGEBRA, 

Ex.  1.   Divide  — -  by 

x'  —  ^a'      x  —  2a       a^  -  Aa"  2a 

X 


ax  -\-  2a*  2a  ax  +  2a'      x  —  2a 

^  (x  +  2a)  (x  -  2a) 

a(x  +  2a) 
=  2,  Quotient. 

Ex  2      ^-^    X    ^'''~^^'     •  ^^~^^' 
'  xCx  +  l)       aj'  +  x  +  l   '  (x  +  1)' 

^(x-l)(x'  +  x  +  l)       (x'-l)(a:'-l)       (x  +  l)' 
x(ix  +  l)  x'-hx-^l  (x-iy 

(x  +  iy    ^    ,, 

=  -^^ ^  J  Result, 

z 


EXERCISE  47. 

Simplify — 

*  14a'c       ISay*'  *  2x  +  2       ic'-xy' 

*  13z»     '    39z**  '     4a'- 1      '   2a  + 1 
126      35a6      _5_^  ^   a;' -  9  .   x-3 

*  25a        48        76''  '  x' i- x  '  x'-l 
9a^       28ax^  _  21a^  (a-l)»  x  +  l 

'   Sc'x        156V    '    106c»*  'a(a;  +  l)'       (a-1)' 

.n  ^  Af\Ji  Q/*.2 1 


492/"                      40ar^  9a;' -1      12a; -18 

15x            2x(a;+l)  ^^    2a;'-a;-l  w4a;'-l 

'  2a;(2x-l)           5ar^  '2x'  +  x-l        x^-l 

13     Q^y  —  ax^y  .    a'y  — 2aa;3/4-a;'2/^ 

aV  -f  a'a;'2/  a'  -f  ay 


FRACTIONS,  136 


14. 


(a  +  1)'      (a  +  1)' 
_     Z7?  +  x-2    ^  ear' 


16. 


4a:'-42;-3  2x'-a;-3 
2a;'  — a;  — 6  ,  2a:^  +  x-3 
2x'  +  a;-l  *  2x'  +  3x-2 
2a;- 2 


17.    a;+ X 


\        x-lj       ar'-fl 

6x2/       ^  +  y     (p^—yy     2a^ 

*  4(a  +  6)'       9(a-6)'    "  8(a  +  6)* 

21  a:'  +  2a;-3  ,,  a:'  +  2x-15    .  ar»  +  Sx* 

*  a:"  4- a; -12        a;'  +  2a;-3     '  a:'  +  4a;'* 

22  6x^-4x3/'       30a: +  203/         ^V 

'   Adx"  ~  20y'  4xy  x  +  y 

23  ^^'-5a:-4        6a;'  +  a;-2        2a;'  +  5a;-12 

*  2x'  +  7a;-4       4x'-4a;-3        9a;'-6a;-8 

-^•)(-i)x('-5^)- 

26.    .   ^  +  ^    -X      ^-^      X 


a:*  +  a;'3/  -f  xi/'      x^  —  xy  -{-  y'^ 


\        a;-3// 


27    a:^-(a-iy      (g  +  xy-l   .  a  +  x-l 

*   a'-(a;  +  l)'      l-(a-x)'  *  a-x-1 

23^2-6-a^^-6'-46-4_^^- 


6-2-a      a'-h6'  +  2a6-4      6'-a'  +  4a-4 


136  ALGEBRA. 

12x'-xy-20y'       27^ -]- bxy-12y'     Gar*  +  2Bx'y  +  Ibxy^ ^ 
'  12x'-Sxy-15y'      3x' -^  bxy  -  12y'      ix"  +  21xy  +  20y' ' 

\ab      be       acj  \b      c      a  J  a'bV 

\a^       TT       ax        /       L         ax  x         J 

rw,  -\-2n      m  —  2n~|  ^  rm  +  2n  _  m  — 2n~| 
Lm  —  2n      m  +  2nJ      Lm  —  2n      m  +  2nJ 

IV.  Reduction  of  Complex  Fractions. 

147.  A  Complex  Fraction  is  one  having  a  fraction  in  its 
numerator  or  in  its  denominator,  or  in  both. 

In  simplifying  any  complex  fraction  it  is  important  to 
write  down  the  entire  fraction  at  each  step  of  the  process. 


Ex.  1.   Simplify 


a:X-^— =  -^-,  Remit. 


y       y 

If  the  numerator  and  denominator  of  the  complex  fraction 
each  contain  fractions,  the  simplification  is  often  effected  most 
readily  by  multiplying  both  the  numerator  and  denominator 
by  the  lowest  common  denominator  of  the  fractions  contained 
in  them. 

Ex.  2.  Simplify    "     V     ^  . 
y      %      X 


FRACTIONS.  137 

Multiply  both  numerator  and   denominator  by  xyz^  and 
obtain 

«  z  -f  x'if  H-  yz^ 

Ex.  3.  Simplify  


.+-i 


cc-2 


A  fraction  of  this  form  is  called  a  continued  fraction.  In 
simplifying  a  continued  fraction  begin  at  the  bottom,  and 
reduce  by  alternate  conversions  of  a  mixed  quantity  into  an 
improper  fraction,  and  divisions  of  a  numerator  by  a  frac- 
tional denominator.     Thus, 

1 1_ 1 

X  H X  -\ X  -+ 


._     3  a;  — 5  a;-5 


a;-2  x-2 

1  x-5 


Simplify — 


x'-Ax-2       x'  -4x-2 
x-5 


EXERCISE  48. 


)  Result, 


4  1  .  1 


X  X 

X  _  a; 


1. 3.  T'  ^' 


a  +  1 


X 


1  .    ,       1 


14--  1--  1  + 

2  re  a-1 

2-i  ^-2c£  6.1+       ^ 


«  c 


2.  ^-  4.-^^ 1-- 

,       1  2cd  a 


138  ALGEBRA. 


14. 


7. 

8. 

U    yJ  . 

9. 

o          =t-l 

-f 

10. 

-lil- 

o  — 1 

11. 

3              ^ 

"     1  +  a 

4       |a-l 

12. 

a  — 1           a 

1          1 

a  —  1       a 

1Q 

13        2 

X       x'        x' 

I-' 

^-1+^ 


15.  1 


l  +  a 


1-a 


16. 

X        1    1  ~* 

l  +  a;          x 

X           \—x 

1  +  X               X 

17. 

2^  +  l_^ 
3/              a; 

18. 

e{a^hy-a^h^ 

a'bV 

io 

a      a:              ax 

ly. 

^_a_2       1 
a      a;      a      aa; 

20. 

-2). 

2x-l        ^ 


FRACTIONS. 

22.  2a- 1 


139 


a-1 


1  +  a 


23. 


24. 


25. 


26. 


27. 


28. 


29. 


l-jjah-cdf 
(ab-iy-c'd' 
(cd  +  iy-a'b' 
(ab  +  cdy-l 


1 ]_    ^i^+        2bc       )• 


a      6  +  c 
1 


a;  — 


«  + 


1 


X 


x  +  2 

l+ar« 


a:  +  : 


a;-2 

l-a:» 


x^ 


1  + 


1  + 


1-a; 


1- 


1  +  x 


xf  +  T^y 


'  x^  -\r  o^y  -\-  7^y* 
(x^-yy 


M 

j_/i+iY  i-/i-iY  /i+i-^y 

a" \b cl       b'       \a      cf       \a      c       b) 

b'      \a      c]       e       \a      b)      [a      c)       b* 


1-2 


1+2 


l-2a; 

l+2a;         4a-^  +  a^)-i 

14- 2a;         ii^  +  x  +  ix')-^ 

\-2x 


140  ALGEBRA. 

EXERCISE  49. 

REVIEW. 

Reduce  to  their  simplest  form — 

1       o^-^x  +  2      .  g  1  -  iri  -  3(1  -  x)^ 

'■----  •  l-|[l-2(l-a;)]' 

7   2(^_li)_i(^_zi). 
'  3(2;  -  I)      ^{x  +  I) 

8.3- 1 

2x 


3. 


x' 

+  a;2- 

-3a; -2 

x' 

-  2x'^  +  4a;  -  3 

x"- 

-5x-2 

+  13a;  -  9 

Sx 

-4 

3a; -2 

2x  +  d 

2a; +  1 

Xj- 

-2n 

2ri  +  a; . 

3..-     3x 


a;  +  1 


4  •«-  ~  *^"'  4-  "^'-  ■  -^ .  9    2  —  a;  _  a;  +  2  ^      6a;     . 

'  a;  +  2n      2n  -  X  *  1  -  2a;      2a;  +  1       4a;''  -  1 

|(|-a;^  +  ia;-2).  ^(^  +  2;^  -  V) 

Kfa;^  +  fa;  -  1)  10.  ^-^f- — ^. 

»^2  2  ^^  x^  -^  xy  —  2y^ 

11       1      +      2  3       ,  4a;  -  3         _ 


a;  -  1      a;  -  2      a;  -  3      (a;^  -  a;)  (a;  -  2) 

a--         a  +  -        a-^      a6  +  - 
a  a  a'  a 

13   L/l--L\  +_A_  _6r2(a-6)_:^) 
2  U      a  -  6  /       a  +  6      2  \      a^  -  6^       / 

1.1  1 


14. 
15. 


9a;2  +  9a;  +  2        1  -  9a;2       4  -  9a;2       2  -  9a;  +  9a? 

3 ^ 2_ ^ 1 

a;2  -  3a;  +  2       (a;  -  1)  (3  -  x)        (2  -  x)  {x  -  3) 


16.  2a; +  y  _  1 ^_  ^      a^'     . 

aJ  +  2/  y  —  X      y^  —  x^ 

17.  («  +  ?  _  2)  («  +  ^  +  2U  (^  -  ^y. 
\a;      a        /  \x      a        I       \x      a} 

18    i  1  +  a;  _  1  +  a;^  ].  ^  /  L±  ^'  _  L±^  1. 


FRACTIONS. 


141 


19. 


20. 


21. 


22. 


23. 


24. 


25. 


\  a  /         ^  ax  I 


x-4- 


X 


x-2 


-2 


x-4 
2a  +  3 


a;  —  5 
3a  +  2 


2a2  +  a  -  1      3a'  +  a  -  2 


6 


(a +  1)2       2-7a  +  6a2 


(I -!)(!-)• 


1  -  ^ 

a;2       9 


(^-f)(^^a)         i'-t) 


o      a 

1  +  8a:' 


1 


9l 
1  -  27a:' 


2a; 

1  + 


2a; 


1  + 


3a; 


3a; 


l-2a; 


5x^  +  4       XX 
X  


x^  +  1 


1  +  3a; 


2+i 


1  +  ^ 
'>•« 


27. 


28. 


/      ^-1  +  2/      \ 

/^  _  2  +  ^K  (    ^'    \  x\—^ ^— I 

\y^  xV       \xy  +  y^)      \  a^  -  2x^y  +  xy^  / 

\a~b) 


i_6  +  6! 


6> 


1  +  6  +  6; 


1.^^ 


142 


ALGEBRA. 


1  + 


29. 

30. 

31. 

32. 
33. 
34. 

35. 


x" 


^-h 


X 


X  + 


('-9 


l)  +  i 


\{x 
X  — 


X  + 


K^  +  i)  -  fa; 
1       .  o         _,      „  .       7 


a;  +  1 


+  3 


X  + 


X  +  1 


+  1 


x-1 


+  X  +  S 


1  + 


a'  -  a;'  +  6a  +  9  ^   a^  -  x^  +  Gx  -  9 


x-1 
x^  —  Sx  +  ax 


X-  —  a'  —  6a 

M^a  -  i[2a 
2a 


3i  -  Y  +  |{f«  -  |[f 


9      9  -  a»  -  a;2  +  2aa;      a^  -  3a 
|(3a  +  2(a  -  5))]  +  2  +  ^a}. 

i(4a-4(5a-3))]}. 


ax 


1  + 


1  + 


1  +  x 
1  -  3a; 


1  -3 


1  +  x 
i-Sx 


1  + 


1+x 
l-3x 


36.  2  + 


^1  +  X 
1  -3a; 

1  + 
I 


4  + 


2  + 


3  -  2a 
a-1 


37. 


|1  4-  an 
1-a 


1  +  aJ 


38. 


39. 


40. 


1  + 


1  n 


1-1 
a; 

/g  +  bey 
\a  —  6c/ 


X 

a  +  be 
a  -be 


-1 


f — ^^^^1 


1 

x_ 

1  + 


a  J 


^-1 


+  1 


(g  +  bey  _  g  -f  6c  ,  ^ 
g  —  6c/       a  —  be 


a  +  l 


a-1 


+  1 


m)'^mr-~{-y 


CHAPTER    XI. 
FRACTIONAL  AND   LITERAL  EQUATIONS. 

148.  General  Method  of  Solution.  If  an  equation  con- 
tain fractions,  it  is  necessary  to  clear  the  equation  of  fractions 
before  transposing  terms  and  solving  by  the  method  given  in 
Chapter  VI. 

Ex.  1.  Solve       X  -\ =  6  H 

6  4 

Multiply  both  members  of  the  equation  by  12,  the  L.  C.  D.  of  the 
fractions. 

12a;  +  2(x  +  2)  =  60  +  S{x  -  4) 
12a;  +  2a;  +  4  =  60  +  3a;  -  12 
12a;  +  2a;  -  3a;  =  60  -  4  -  12 
11a;  =  44 
a;  =  4,  Root 

If  a  fraction  be  preceded  by  a  minus  sign,  it  is  important  to  remember 
that  the  sign  of  each  term  of  the  numerator  must  be  changed  on  clearing 
the  equation  of  fractions. 

^     ^    ^,       a;  +  l       2x-5       lla;  +  5       x-1^ 

Ex.  2.   Solve —  =  — -— — • 

2  5  10  3 

Multiplying  by  30, 

15a;  +  15  -  12a;  +  30  =  33a;  +  15  -  10a;  +  130 
16a;  -  12a;  -  33a;  +  10a;  =  -  15  -  30  +  15  +  130 
-  20a;  =  100 

«  =-  -  5,  Boot. 

lis 


144  ALGEBRA, 

Ex.  3.   Solve  --^  +  ^^  -  ^^  =0. 

l-\-x       1—x       1  —  ic' 

Multiplying  by  the  L.  C.  D.,  1  -  x^, 

4(1  -  a:)  +  (a;  +  1)2  -  a;2  +  3  =  0 
4-4a;  +  a;2  +  2a;  +  l-a;2  +  3  =  0 

-  2a;  =  —  8 
X  =  4,  -Koo^ 
Hence,  in  general, 

Clear  the  equation  of  fractions  by  multiplying  each  term  hy  the 
L.  C.  D.  of  all  the  fractions ; 

For  a  fraction  preceded  hy  the  minus  sign  the  sign  of  each  term 
of  the  numerator  must  be  changed  when  the  denominator  is  re- 
moved ; 

Complete  the  solution  by  the  methods  of  Chapter  VI. 

149.  Equations  involving  Decimal  Fractions.  If  one  or 
more  of  the  coefficients  of  an  equation  is  a  decimal  fraction, 
we  may  solve  the  equation  by  expressing  the  decimal  frac- 
tions as  common  fractions. 

If  all  the  coefficients  are  decimals  or  whole  numbers,  it  is 
simpler  to  solve  directly  in  decimals. 

Ex.   Solve  .3x-.14a;  =  .012x  +  .592. 

.32;  -  .14a;  -  .012a;  =  .592 
.148a;  =  .592 
a;  =  4,  Boot. 

EXERCISE   50. 

Solve — 

3^      5x_2^      11  Sx      2x  +  7  _  4x-\-5  ^ 

*4        6~3        2*  '5            3      "15* 

X      3a;      7a;  _  34  K^t^_,^_*|4'^ 

'  Z~J      y~15*  '  2      3      5~4      "6* 

2x-3       :MJ.  ^  5x  +  2  i__?4.A  =  A_?:?. 

'       4  6     ~     12  '  2a;      cc      3a;  ~  4*      24* 


FRACTIONAL  AND  LITERAL  EQUATIONS.  140 

7  3  2- 


■«-!^|- 

16.  f(x-l)=K^-2). 

"•M-- 

17.  i(l  +  a;)=i(2x  +  l). 

12.  -  +  2  =  0. 

X 

18.  3(|a;-l)(ia;  +  |)=a;», 

10  ^-^ 

2a;  4- 3 

6-5a; 
12 

^^-      3 

6 

-^- 

x-Z 
6 

-1^'^  +  S^ 
3          24 

oi    3x-l 

a;  +  l 

4a:  +  l       3(a:-l)      ^ 

^''       7 

6 

21-4           ^- 

22.  10--?^^-3a;-4i  =  0. 
4 

2a: +  5       ^+li  ,^_5£:zM      1 
_^     cc  — 1       23;-f3    ,  a;  a; 

^-  ^"^^  +  ^)  +  ^  -  K2^  +  6)  =  ^^ 

,.  "        ic  +  S       a;  +  7    ,    a;  4- 1       2a;-5       a;  +  22 

^  ,  27.  — ' h = —• 

7  6  2  10  70 

10 


146  ALGEBRA. 

28    2a;  +  3       5a;  4- 1        3a;+l  ^1       6a;-l       a;-f9 
5  6  4  3  15  30   ' 

29.  |(5a;  +  2)  -  |(7a;  -  2)  +  i(3a;  -  2)  -  a;  -  -• 

80    y  +  6       2y-18       2i/  +  3^  3y  +  4 

■      11  3  4  ^         12     " 

32.0.5.-0.4.  =  0.3.  "-    1-5--1-6      3.5.-2.4 

33.  1.5a;  — 5  =  a;. 

34.  1.25a;  +  1.9  = -1.125a;, 

35.  0.6a;  -  1.5  =  0.2  -  0.15a;.     39. 


,  2  +  -^==^-M 
1.2      3        9 


1.2 

0.8 

9Q 

3.2a; -3.4      0.6a;  +  4 

4.5 

2.5 

39. 

.0032a;- 
0.1a; 

- 1        1.005 
.0125a; 

40, 

.0001  _ 

1.00005 

80a; 

200.01 

''■          OA 

42. 

a;-l 

d;  +  l 

3 
5* 

43. 

5 
2x-l 

8 

3a; +  1 

44. 

2a; +  5 
5a; +  3 

_  2a;  +  l 
5a; +2 

45. 

2a;-5 

2a; +  2 

3a;-5 
3a;-3 

46. 

6a;-5 
3a:-3 

8a; -7 
4a;  +  4 

0.6a;  +  .045       5a;  -  1.78  ^  ^^^5 


X      a;'  -  5a;  _  2 
'  3       3a;-7~3' 

48.-A_+-l-^-^^ 
1-a;       1  +  a;       1  — a;' 

3-a;       34-a;       9- a;' 
2a;  4-1  10  2a;-l 


50. 


2a;- 1      4a;' -1      2a; +  1 


51.^+      '  ' 


x  +  1       x-1       x  +  2 


62. 


FRACTIONAL  AND  LITERAL  EQUATIONS,  147 

x-1^     3  l-3a; 

ar'-S      x-2       x^-\-2x-^4' 


^^    a^-x-^1       _        x'  +  aj  +  l 
63. =  2x 


64. 


x-1  x+1 

a;-3       jx'  +  l^   x  +  S 
2x4-6       x^-d       Sx-d 


55.  ^  +  -^-..^  =  .^-1- 


66. 


x-1      x  +  1       2x-2       3a;H-3      l-x» 
x  +  l        cc'  +  T  ^      2  x-1 

2x-3      4x^-9      2x  +  3      6-4x* 


67. — 2  —  X  = 


3x-6      6x  +  12  2x^-8 

4  2  3 

68.  — + r : r  + 


3x'  +  9x  +  6       x'-3x       x^-x-6      7?-2x-Z 


4 


3x'  +  6x 

x  +  1 


'  x'  +  x-6       3x-x'-2       x=^  +  2x 


60. 


6x  +  6  2x  +  1  2x 


2x^  +  5x  +  3      2x^-x-l      x'-f-2x 


150.  Special  Methods.  The  work  of  solving  an  equation 
may  frequently  be  diminished  by  using  some  special  method 
or  device  adapted  to  the  peculiarities  of  the  given  equation. 

1st  Special  Method.  If  the  denominators  of  some  fractions 
are  monomials,  and  of  some  are  polynomials,  it  is  best  to  make 
two  steps  of  the  process  of  clearing  the  equation  of  fractions, 
the  first  step  being  to  remove  the  monomial  denominators 
and  simplify  as  far  as  possible  before  proceeding  to  the  second 
step,  which  is  to  remove  the  remaining  polynomial  denomi- 
nators. 


148  ALGEBRA. 

^      ,     ^,        2a; +  81        13a;- 2     ,  a;      7x       a;  + 16 

Ex.  1.   Solve -^ h  -  = 

9  17a; -32       3      12  36 

Multiplying  by  36,  the  L.  C.  D.  of  the  monomial  denominators, 

8a;  +  34  -  36(13a;  -  2)  ^  -^^^  =  21a;  -  a;  -  16. 
17x  -  32 

Transposing  all  terms  except  the  fraction  to  right-hand  side, 

36(13a;  -  2)  ,_ 
17a;  -  32 

Dividing  by -2,        M^^:^  =  25 

234a;  -  36  =  425a;  -  800 
191a;  =  764 
a;  =  4,  Root. 

2d  Special  Method.  Before  clearing  an  equation  of  frac^ 
tions  it  is  often  best  to  combine  some  of  the  fractions  into  a 
single  fraction. 

x  —  1       x  —  2       x  —  Z       x~4: 


Ex.  2.   Solve 


a;  —  2       a;  —  3       a;  —  4       x  —  5 


In  this  equation  it  is  best  to  combine  the  fractions  in  the  left-hand  mem- 
ber, and  those  in  the  right-hand  member,  each  into  a  single  fraction,  before 
clearing  of  fractions.    We  obtain 


{x  -  2)  (a;  -  3)      {x  -  4)  (a;  -  6) 


Clearing  and  solving,  x  ^  -' 

,2t 


Solve — 


EXERCISE  51, 


3a; -1  4x      _  a;  +  5 

6  3a;  +  2  ~      2     ' 

3-2a;  x       l-6a;  ^  2-3g 

'       4  6      15-7a;  9 


1.2. 


13. 


FRACTIONAL  AND  LtTEttAL  EQUATIONS. 
3  2x-l      X 


14d 


3.  21- 


2a: +  4 


6x  +  13      2a;  +  5 


23 


12 
5 


6 
2ia;-3 


4x-36 
a; +  11 


5-^a; 


8 


3a;-l       4x-7 


7. 


30 
6a;-7 
11a; +  5 

a;  +  4 


15 

a;  +  l 
15 


2a; 


■    lla;  +  5 
16 

3        7x 


0. 
15 


+ 


8.  3  + 


7a; +  11 
3x-2|       7 


10. 


11. 


I 


9 

5 -4a; 


^-n 2_ 

12      |a;  +  ll 


12a; -11 
2a;  -  1  ^  199 
30      ~  'lO  ' 
3x       4a; +  9 

12 
-i       2x-li 


60 


4-a;      5 
24        4 


2a;  +  3i 


6 


3x-l  +  iO 

2r2x      lf3a;-l 
3L9       2I      6 


/2ix  +  l\      3 
Vila; +  6/      2 

11 


O.la;- 100.01  ^  10 
1.001a; +  .0002  .05 
1  -  1.4a;  _  0.7(a;  +  1) 


2a;-5 

7a; +  8 

0.2a; 


=  l-x. 

19a; +  3 


14. 


-1.6 


54 
2a; -0.2 


=  0. 


16. 


17. 


18. 


19. 


2  +  a; 
1.2a; 


1.5 

a; -0.25 
0.125 

1 


l-0.5a; 
1.5        0.4x+l 


15. 


0.4 

2a;-3 

0.3a;  -  0.4 

0.4x  + 1 


4a;  +  0.5  "^  2 
0.4X-0.9 


.06*-. 07 


0.2a; -0.2 
0.4a;  - 1       5a; 


0.5 
20 


.05 


3x 


1 


0.2 

1 


a;-2  a;-3 
a;-l  a;-3 
x-2      x-4 


x-4 
x-5 


x-5 
x-7 


x-Q      x-S 


150  ALGEBRA, 

20 


x  —  A 
8       a;-9       a:-5       x~Q 


21.—^—  2  2  ^ 


3x-2        2a;-3       2a;  +  3       3a;  +  2 

^„^  2a;  +  l     ,   2a:4-9       2a;  +  3        2:c  +  7 

x  +  1     .    a;  +  5         a;  +  2         a;  +  4 

4x-17       10a;-13       8a;  -  30       5a;-4_Q 
a;-4  2a;-3  2a;-7         a;-l    ~    ' 

151.  Literal  Equations  are  equations  in  which  some  or  all 
of  the  known  quantities  are  denoted  by  letters ;  as,  a,  6,  c  .  .  ., 
or  m,  n,  ^  .  .  . 

The  methods  used  in  solving  literal  equations  are  the  sama 
as  those  employed  in  numerical  equations. 

Ex.  1.   Solve  a{x  —a)  =  b(x  -b). 

ax  —  a^  -=  bx  —  b' 
ax  -bx  =  a^  —  b^ 
{a-  b)x  =  a^-b^ 
X  =  a  +  b. 

Ex.2.  Solve  ^I1*  =  ^±A. 
x~c      x-\-2c 

{a  -  6)  (a;  +  2c)  =  («  +  6)  (a;  -  c) 
ax  +  2ac  —bx  —  2bc  =  ax  +  bx  -  ac  —  bo 
-  2bx  =  -  Sac  +  be 

26 

EXERCISE   52. 

Solve — 

1.  3x  +  2a  =  a;  +  8a.  ^,  bax  —  c  =  az  —  bc. 

2.  9aa;  — 36  =  2ax  +  46.  4.  aa;  +  6  =  5a;  +  26. 

*  Transpose  the  second  and  third  fractions. 


FRACTIONAL  AND  LITERAL  EQUATIONS.  161 

14. 


5.  Scx  =  a—(2b  —  a  f  ex). 

6.  5x  —  2ax^=S~b.  ^^'   a  —  2x  ~  a-\-  2x 

7.  2ax~Sb  =  cx  +  2d. 

8.  (x4-a)(x-6)=xl 

9.  ab(x  +  1)  =  a" -j- b'x. 
10.  (a;-l)(a;-2)  =  (a;-a)'. 


■'•i(!-')=ie-)- 

12.  (a  — 5>= «'-(«  + 5)a;.         "  2a;-a  a;-a" 

0        a       0      a  a      6      c 

19.  2i:^* +  *£:!-« +  E^  =  o. 

a6  be  ac 

on       ^^      r       ^^  3a'x  +  6' 


21. 
22. 


3a  +  5      3a-6       9a,^-6' 

a;         ^      _     1  a; 

a      a  —b       a  +  b       b 

a^  —  x      b^  —  x      &  —  X  _a^  _  6^ 
e  a  b  e        a 


23   5a'-7a;      a5' +  lOx ^  10c' +  3a;      5(a-c)      6* 
3a6  5ac      ~      66c  36  5c 


a  + 

■  X 

1-? 

— 

a 

a 

24. 

— 

X  — 

a 

1 

X 

a 

a 

1 

X 

b 

25. 

■==■  ■ 

b 

c 

26. 


a-\-bx  a-  hx 


27. 


36  +  2ax  6-2ax 

x  —  1  X 

a       _  o  —  1  ^ 
g  + 1  g 


152  ALGEBRA. 


a-1 

1 

a;  +  l 

1  +  ^ 
a 

31.  a+  -^  =  1  +  -^ 

29.  -  = '—  1  +  -  1-i 

a;  f.      a;                                        a                     a 

I ^~T 

a  0 

152.  Problems  involving  Simple  Equations  containing 
Fractions. 

Ex.  1.  A  has  8|  dollars  more  than  |  as  much  as  B  has, 
and  together  they  have  56J  dollars.     How  many  has  each? 

Let  X  =  the  number  of  dollars  B  has. 

Then    fa;  +  8|  =  "  "        A  has, 

and       f  a;  +  8f  =  "  "        both  have. 

5£      35  _  225. 
•'344* 

20a:  +  105  =--  675 
Hence,  x  =  28|,  the  number  of  dollars  B  has, 

and       fa;  +  8|  =  27f,  "  "        A  has. 

Ex.  2.   Divide  the  number  100  into  two  such  parts  that  the 
eighth  of  the  larger  part  exceeds  the  eleventh  of  the  less  part  by  3. 
Let  a;  =  the  larger  part. 

Then  100  —  a;  =  the  less  part. 

-  =  the  eighth  of  the  larger. 
8 

^^^-^  =  the  eleventh  of  the  less, 
11 


X 

8  ' 

100  -  a;      o 
~       11           ^ 

11a;- 

-  800  +  8a;  =  264 

19a;  =  1064 

X  =  56,  the  greater  part 

100  -  a;  =  44,  the  less  part. 

FRACTIONAL  AND  LITERAL  EQUATIONS.  153 

EXERCISE   53. 

1.  The  sum  of  the  third  and  fourth  parts  of  a  number  is 
14.     Find  the  number. 

2.  Find  that  number  whose  fifth  and  sixth  parts  together 
are  16^. 

3.  What  is  that  number  whose  third,  fourth,  and  fifth  parts 
are  together  13  less  than  the  number  itself? 

4.  The  difierence  between  the  seventh  and  third  parts  of  a 
number  is  5  more  than  one  ninth  of  the  number.     Find  it. 

5.  There  are  two  consecutive  numbers  such  that  one  sev- 
enth of  the  greater  exceeds  one  ninth  of  the  less  by  one. 
Find  them. 

6.  There  are  three  consecutive  numbers,  such  that  if  the 
first  be  divided  by  6,  the  next  by  7,  and  the  largest  by  8,  the 
sum  of  the  three  quotients  is  \  more  than  ij-  of  the  sum  of  the 
three  numbers.     Find  them. 

7.  The  difference  of  two  numbers  is  9,  and  -j^  of  the  less, 
increased  by  3,  is  ^  of  the. greater.     Find  the  numbers. 

8.  A  man  left  half  his  property  to  his  wife,  one  fifth  to  his 
children,  a  twelfth  to  a  friend,  and  the  remainder,  $2600,  to  a 
hospital.     How  much  property  had  he  ? 

9.  In  a  certain  orchard  there  are  apple,  pear,  and  cherry 
trees :  ten  less  than  one  half  are  apple,  twelve  more  than  one 
third  are  pear,  and  four  more  than  an  eighth  are  cherry  trees. 
How  many  trees  are  there? 

10.  Find  three  consecutive  numbers,  such  that  if  they  are 
divided  by  2,  3,  and  4  respectively,  the  sum  of  the  quotients 
will  be  the  next  higher  number. 

11.  Divide  $130  among  A,  B,  and  C,  so  that  A  receives  J  as 
much  as  B,  and  C,  f  as  much  as  A  and  B  together. 

12.  The  sum  of  two  numbers  is  97,  and  if  the  greater  be 
divided  by  the  less,  the  quotient  is  5  and  the  remainder  L 
Find  the  numbers. 

Hint.  The  divisor  multiplied  by  the  quotient  is  equal  to  the  dividend 
diminished  by  the  remainder. 


154 


ALGEBRA. 


13.  Divide  the  number  100  into  two  such  parts  that  the 
greater  part  will  contain  the  less  3  times  with  a  remainder 
of  16. 

14.  The  difference  between  two  numbers  is  40,  and  the  less 
is  contained  in  the  greater  3  times  with  a  remainder  of  12. 
Find  the  numbers. 

15.  Five  years  hence  a  boy  will  be  f  as  old  as  he  was  3 
years  ago.     How  old  is  he  now? 

16.  A's  age  is  f  of  B's  age,  and  in  7  years  he  will  be  f  as 
old  as  B.     How  old  is  each? 

17.  Eight  years  ago  a  father  was  2>\  times  as  old  as  his  son, 
and  1  year  hence  he  will  be  2^  times  as  old.  How  old  is 
each  now? 

18.  A  tax  of  S5000  was  paid  by  four  men,  A,  B,  C,  and  D, 
A  paying  f  as  much  as  B,  C  half  as  much  as  A  and  B  to- 
gether, and  D  $400  less  than  A  and  B  together.  How  much 
did  each  pay? 

19.  A  man  sold  4  acres  more  than  f  of  his  farm,  and  had  6 
acres  less  than  f  of  it  left.     How  many  acres  had  he  ? 

20.  Find  two  consecutive  numbers,  such  that  f  of  the  less 
exceeds  f  of  the  greater  by  \  of  the  greater. 

21.  If  a  boy  can  do  a  pieoe  of  work  in  15  da3^s  which  a 
man  can  do  in  9  days,  how  long  would  it  take  both  working 
together  ? 


Solution.    Let 
Then 
But 


X  =  number  of  days  both  require. 

-  =  the  part  they  both  can  do  in  1  day. 


And      xV  +  i 
Hence,  xV  +  \ 


the  boy 
the  man 
both 

Or,  X  =  5f. 


Therefore,  they  both  together  require  5f  days. 


These  are  called  common-time  examples. 


FRACTIONAL  AND  LITERAL  EQUATIONS.  155 

22.  A  can  accomplish  a  piece  of  work  in  6  days,  and  B  can 
do  the  same  in  8  days.  How  long  will  it  take  them  together 
to  do  the  work  ? 

23.  A  can  spade  a  garden  in  3  days,  B  in  4  days,  and  C  in 
6  days.     How  long  will  they  require  working  together? 

24.  A  and  B  can  together  mow  a  field  in  4  days,  but  A  alone 
could  do  it  in  12  days.     In  how  many  days  can  B  mow  it? 

25.  If  A,  B,  and  C  can  together  do  a  certain  amount  of 
work  in  5^  days,  which  B  alone  could  do  in  24  days,  or  C  in 
16  days,  how  long  would  A  require? 

26.  A  and  B  together  can  dig  a  certain  ditch  in  1^  days ;  A 
and  C  in  2  days,  but  A  alone  in  3  days.  How  many  days 
would  it  take  B  and  C  together  to  dig  it? 

27.  A  and  B  in  b\  days  accomplish  a  piece  of  work  which 
A  and  C  can  do  in  6  days  or  B  and  C,  in  7^  days.  If  they  all 
work  together,  how  many  days  will  they  require  to  do  the 
same  work  ? 

28.  Two  inflowing  pipes '  can  fill  a  cistern  in  27  and  54 
minutes  respectively,  and  an  outflowing  pipe  can  empty  it 
in  36  minutes.  All  pipes  are  open  and  the  cistern  is  empty ; 
in  how  many  minutes  will  it  be  full  ? 

Hint.  Since  emptying  is  the  opposite  of  filling,  we  may  consider  that 
a  pipe  which  empties  ^^  of  a  cistern  in  a  minute  will  fill  —  3*5  of  it  each 
minute. 

29.  A  tank  has  four  pipes  attached,  two  filling  and  two 
emptying.  The  first  two  can  fill  it  in  40  and  64  minutes 
respectively,  and  the  other  two  can  empty  it  in  48  and  72 
minutes  respectively.  Tf  the  tank  is  empty  and  the  pipes  all 
open,  in  how  many  minutes  will  it  be  full? 

30.  A  man  labors  8  days  upon  a  piece  of  work  which  he 
could  complete  in  4  more  days,  but  he  is  then  joined  by  a 
boy,  and  they  finish  it  in  2|-  days.  In  how  many  days  could 
the  boy  do  the  entire  task  ? 

31.  At  what  time  between  3  and  4  o'clock  are  the  hands  of 
a  watch  pointing  in  opposite  directions  ? 


166  ALGEBRA. 

Solution.  At  3  o'clock  the  minute-hand  is  15  minute-spaces  behind  the 
hour-hand,  and  finally  is  30  spaces  in  advance :  therefore  the  minute-hand 
moves  over  45  spaces  more  than  the  hour-hand. 

Let  X  =  the  number  of  spaces  the  minute-hand  moves. 

Then  a; -45=    "        "        ''       «       '^    hour-hand  " 

But  the  minute-hand  moves  12  times  as  fast  as  the  hour-hand ; 

hence,  x  =  12(a;  -  45).    Solving,  x  =  49xV. 
Thus  the  required  time  is  49x^r  ^^^-  P^st  3. 

32.  When  are  the  hands  of  a  clock  pointing  in  opposite 
directions  between  4  and  5?     Between  1  and  2? 

33.  What  is  the  time  when  the  hands  of  a  clock  are  together 
between  6  and  7  ?    Between  10  and  11  ? 

34.  At  what  instants  are  the  hands  of  a  watch  at  right 
angles  between  4  and  5  o'clock?    Between  7  and  8? 

35.  A  courier  travels  5  miles  an  hour  for  6  hours,  when 
another  follows  him  at  the  rate  of  7  miles  an  hour.  In  how 
many  hours  will  the  second  overtake  the  first  ? 

Solution.     Let  x  =  the  number  of  hours  the  second  travels. 
Then  a:  +  6=    "  "       ''      "         "  first  " 

b{x  +  6)  =    "  "       "  miles     "    "  " 

7x  =   "  "      "      "         "  second        " 

They  travel  equal  distances, 

hence,  7a;  =  5(a;  +  6).     Solving,  a;  =  15. 
Therefore  the  second  courier  requires  15  hours. 

36.  A  courier  who  travels  5^  miles  an  hour  was  followed 
after  8  hours  by  another,  who  went  7^  miles  an  hour.  In  how 
many  hours  will  the  second  overtake  the  first  ? 

37.  A  messenger  rides  for  6  hours  at  the  rate  of  13  miles  in 
2  hours,  when  he  is  followed  by  another  at  the  rate  of  8  miles 
an  hour.  How  many  miles  will  each  travel  before  the  first  is 
overtaken  ? 

38.  A  train  running  40  miles  an  hour  left  a  station  45  min- 
utes before  a  second  train  running  45  miles  an  hour.  In  how 
many  hours  will  the  second  train  round  the  first  ? 


FRACTIONAL  AND  LITERAL  EQUATIONS.  157 

39.  An  express  train  which  runs  65  miles  an  hour  leaves 
a  station  4  hours  after  a  freight  traveling  11  miles  an  hour. 
How  many  miles  from  the  station  will  the  express  round  the 
freight? 

40.  A  boy  starts  on  a  bicycle  2^  hours  after  his  sister,  who 
rode  8  miles  an  hour,  and  overtook  her  in  5  hours.  How  fast 
did  he  ride  ? 

41.  A  gentleman  has  10  hours  at  his  disposal.  He  walks 
out  at  the  rate  of  S^  miles  an  hour  and  rides  back  4J-  miles 
an  hour.     How  far  may  he  go? 

42.  A  and  B  start  out  at  the  same  time  from  P  and  Q,  re- 
spectively, 82  miles  apart.  A  walked  7  miles  in  2  hours, 
and  B  10  miles  in  3  hours.  How  far  and  how  long  did  each 
walk  before  coming  together,  if  they  walked  toward  each 
otlier  ?  If  A  walked  toward  Q,  and  B  in  the  same  direction 
from  Q? 

43.  A  hare  takes  7  leaps  while  a  dog  takes  5,  and  5  of  the 
dog's  leaps  are  equal  to  8  of  the  hare's.  The  hare  has  a  start 
of  50  of  her  own  leaps.  How  many  leaps  will  the  dog  take 
to  catch  her? 

Solution.  Let         x  =  the  number  of  leaps  the  dog  takes. 

Then  ix  =       ''  "  "        hare  takes  tn  same  (tm«. 

Also,  let  w  =       "  "  feet  in  1  leap  of  the  hare. 

Then  |n  =       "  "         "  "  dog. 

Hence  a;  x  —  =  -^  =  the  number  of  feet  in  the  whole  distance. 

5  5 

And  (^  +  50 )  X  n  =  ^^  +  50n  =  the  number  of  feet  in  the  whole 
^^  ^  ^  distance. 

Therefore  §!^  =  ZM  +  50n ;  or,  a;  =  250. 

5  5 

Thus  the  dog  will  take  250  leaps. 

44.  A  hare  is  50  leaps  in  advance  of  a  hound,  and  takes  5 
leaps  to  the  hound's  3,  but  2  of  her  leaps  are  equal  to  1  of 
his.  How  many  leaps  must  each  take  before  the  hare  is  caught? 


158  ALGEBRA. 

45.  A  greyhound  pursues  a  fox  which  has  a  start  of  60 
leaps,  and  makes  3  leaps  while  the  greyhound  makes  2. 
Three  of  the  dog's  leaps  are  equivalent  to  7  of  the  fox's. 
How  many  leaps  does  each  take  before  the  hound  catches 
the  fox? 

46.  A  has  a  certain  sum  of  money,  from  which  he  gives  B 
$3  and  \  of  what  remains  ;  he  then  gives  C  $6  and  \  of  what 
remains,  and  finds  that  he  has  given  away  half  his  money. 
How  many  dollars  had  A  ? 

47.  A  colonel  in  arranging  his  troops  in  a  solid  square  found 
he  required  51  men  to  complete  it ;  but  on  making  each  side 
contain  one  man  less,  had  32  men  more  than  the  square 
required.     How  many  men  had  he? 

Hint.     Let  x  =  the  number  of  men  in  each  side  of  his  first  square. 

48.  A  regiment  is  arranged  in  the  form  of  a  hollow  square 
15  men  deep,  containing  1800  men.  How  many  men  on  the 
outer  side  of  the  square? 

49.  An  officer  arranged  his  troops  in  a  rectangle,  with  3 
men  more  on  a  side  than  on  an  end ;  if  he  should  form  a  square 
each  side  of  which  is  equal  to  one  more  than  the  width  of  the 
rectangle,  he  would  have  44  men  left.  How  many  men 
had  he? 

50.  If  a  bushel  of  oats  is  worth  40  cents  and  a  bushel  of 
corn  is  worth  55  cents,  how  many  bushels  of  each  grain  must 
a  miller  use  to  produce  a  mixture  of  100  bushels  worth  48 
cents  a  bushel? 

61.  A  man  has  $5050  invested,  some  at  4%,  and  some  at  5%. 
How  much  has  he  at  each  rate  if  the  annual  income  is  $220? 

52.  Divide  the  number  54  into  4  parts,  such  that  the  first 
increased  by  2,  the  second  diminished  by  2,  the  third  multi- 
plied by  2,  and  the  fourth  divided  by  2,  will  all  produce  equal 
results. 

53.  Two  laborers  are  hired  at  $3  and  $4  a  day  each  ;  together 
they  did  35  days'  work,  and  each  received  the  same  sum.  How 
many  days  was  ^ach  employed? 


FRACTIONAL  AND  LITERAL  EQUATIONS.  159 

64.  Divide  180  into  two  such  parts  that  if  the  less  be  sub- 
tracted from  f  of  the  greater,  the  remainder  is  ^  the  difference 
of  the  two  parts. 

55.  A  boy  bought  some  apples  at  the  rate  of  5  for  2  cents :  he 
sold  J  of  them  at  the  rate  of  2  for  a  cent,  and  the  rest  at  f  of  a 
cent  apiece  ;  he  made  5  cents.    How  many  apples  did  he  have? 

56.  A  lady  in  reading  a  book  read  the  first  day  half  the 
pages  and  1  more ;  the  second  day  half  the  remainder  and 
1  more ;  the  third  day  half  the  rest  of  it  and  1  page  more, 
and  still  had  40  pages  to  read.  How  many  pages  were  there 
in  the  book  ? 

57.  Find  two  numbers  whose  difference  is  12,  such  that  if  ^ 
the  less  be  added  to  ^  the  greater,  the  sum  shall  be  equal  to  J 
the  greater  diminished  by  \  the  less. 

58.  A  man  buys  two  pieces  of  cloth,  one  of  which  contains 
3  yards  more  than  the  other.  For  the  less  piece  he  pays  at 
the  rate  of  $5  for  3  yards;  for  the  other  $1.50  per  yard;  he 
sells  the  whole  at  the  rate  X){  $9  for  5  yards  and  gains  $36. 
How  many  yards  were  there  in  the  less  piece? 

59.  A  and  B  together  can  do  a  piece  of  work  in  2f  days;' 
A  and  C  in  2f  days ;  and  B  and  C  in  2^\  dajrs.  How  many 
days  will  each  require  to  do  the  work  alone  ? 

60.  At  what  instants  are  the  hands  of  a  watch  at  right 
angles  between  10  and  11  o'clock? 

61.  A  fox  is  100  leaps  ahead  of  a  dog  in  pursuit.  The  fox 
makes  3  leaps  in  the  same  time  that  the  dog  makes  2,  but  3 
of  the  dog  are  equal  to  5  of  the  fox.  How  many  leaps  will 
each  take  before  the  fox  is  caught  ? 

62.  A  does  y  of  a  piece  of  work  in  10  days,  when  he»receives 
the  aid  of  B  and  C.  They  finish  it  in  3  days.  If  B  could  do 
the  entire  task  in  30  days,  in  what  time  could  C  do  it  alone  ? 

63.  A  man  has  a  hours  at  his  disposal.  How  far  may  he 
ride  in  a  coach  which  travels  b  miles  an  hour,  and  return 
home  in  time,  walking  c  miles  an  hour? 

64.  Separate  a  into  two  parts  such  that  the  greater  divided 


160  ALGEBRA. 

by  the  less  may  give  b  for  a  quotient  and  c  for  a  remainder. 
Prove  your  result. 

65.  The  fore  wheel  of  a  carriage  is  a  feet  in  circumference 
and  the  hind  wheel  is  b  feet.  What  is  the  distance  passed 
over  when  the  fore  wheel  has  made  c  revolutions  more  than 
the  hind  wheel  ? 

158.  Problems  in  Interest.  The  power  of  algebra  as  com- 
pared with  arithmetic  is  well  illustrated  by  the  algebraic 
treatment  of  questions  relating  to  interest. 

Let  p  =  the  number  of  dollars  in  the  principal, 
r  =  rate  of  interest  expressed  decimally, 
t  =  time  expressed  in  years, 
i  =  the  interest*  on  the  principal  for  the  given  time  and 

rate, 
a  ==  amount,  i.  e.  the  sum  of  the  principal  and  interest. 

154.  I.  Problems  involving"  Interest,  Principal,  Rate, 
Time. 

.  The  principal  multiplied  by  the  rate,  that  is,  pr,  gives  the 
interest  for  one  year,  and  prt  gives  the  interest  for  t  years. 

.  • .    i  =prt (1) 

If  interest,  rate,  and  time  be  given,  to  find  the  principal,  in 
equation  (1)  t,  r,  t,  represent  known  quantities,  and  p  the  un- 
known quantity.    Solving  (1)  for  jp, 

2'  =  4-   •   • (2) 

rt 

In  like  manner,  if  interest,  principal,  and  time  are  given,  to 
find  rate,  solve  (1)  with  reference  to  r, 

r  =  4 (^) 

pt 

So  also  if  interest,  principal,  and  rate  are  given, 

t  =  - (4) 

pr 


FRACTIONAL  AND  LITERAL  EQUATIONS.  161 

Hence,  in  general,  if  any  three  of  the  four  quantities, 
I,  p,  r,  i,  are  given,  the  remaining  quantity  may  be  found  by 
substituting  for  the  three  given  quantities  in  equation  (1), 
and  solving  the  equation  for  the  remaining  unknown  quan- 
tity, or  by  substitution  in  one  of  the  formulas  (2),  (3),  (4). 

Ex.  1.  At  what  rate  will  $50  produce  $20  interest  in  6  years 
and  8  months  ? 

We  have  given  p  =  $50 

i  =  120 
t  =  6f  years, 
to  find  r. 
Using  formula  (3), 

r  =      20      _  20  X  3  _  _3  ^  ^ 

50  X  6f      50  X  20      50 
r  =  6%,  Rale. 

155.  n.  Problems  involving  Amount,  Principal,  Bate, 
and  Time. 

^  Since  amount  =  principal  +  interest, 

a=p-\-prt  .   ,-',-  .  *    i    .    .    .(5) 

Any  three  of  the  four  quantities,  a,  p,  r,  t,  being  given, 
the  remaining  quantity  may  be  found  by  solving  equa- 
tion (5). 

Th-.  P-T^ (6) 

r  =  ^^ (7) 

pt 

t  =  ^^ (8) 

pr 

Thus  all  the  problems  of  arithmetic  relating  to  interest, 
principal,  rate,  time,  and  amount  may  be  solved  by  means 
of  equations  (1)  and  (5). 
11 


162  ALGEBRA. 

Ex.  1.   Find  the  principal  that  will  amount  to  $335.30  in  8 

years  and  6  months  at  5%. 

Here,      given  a  =  335.30,        t  =  8^,        r  =  .05,        find  p. 

TT  •      r         1    /c^      ^  335.30  335.30 

Uemg  formula  (6),    P  =  J^^J^T:^  =  ^m 

-  $235.30,  Principal. 

EXERCISE   54. 

1.  Find  the  interest  of  $650  for  4  years  at  6  per  cent. 

2.  Find  the  time  in  which  $325  will  produce  $84.50  interest 
1 5  per  cent. 

3.  Find  the  rate  at  which  $176  will  yield  $43.56  interest  in 
5  years  6  months. 

4.  What  principal  at  5  per  cent,  will  produce  $102.30  in  7 
years  and  9  months? 

6.  Find  the  time  required  for  $123.45  to  amount  to  $197.52 
at  6  per  cent. 

6.  Find  the  rate  at  which  $75.60  will  amount  to  $91.98  in  5 
years. 

7.  At  what  rate  will  $15  amount  to  $16  in  10  months?      • 

8.  What  principal  will  produce  a  dollar  a  month  at  3|  per 
cent.  ? 

9.  In  what  time  will  the  interest  on  a  sum  of  money  equal 
the  principal  at  4  per  cent.  ?  At  6  per  cent.  ?  At  5 J  per 
cent.  ? 

10.  What  principal  in  2  years  4  months  will  amount  to 
$609.30  at  5^  per  cent.? 

11.  In  what  time  will  $76.80  amount  to  $80  at  ^  per 
cent.? 


CHAPTER    XII. 
SIMULTANEOUS  EQUATIONS. 

156.  Simultaneous  Equations  are  equations  in  which 
more  than  one  unknown  quantity  is  used,  but  in  any  set 
of  which  equations  the  same  symbol  for  an  unknown  quan- 
tity stands  for  the  same  unknown  number. 

Thus,  in  the  group  of  three  simultaneous  equations, 

a;  4-  2/  +  2z  =  13, 

2x  4-  3/  -  z  -  3, 

X  stands  for  the  same  unknown  number  in  all  of  the  three 
equations,  y  for  another  unknown  number,  and  z  for  still 
another. 

167.  Independent  Equations  are  those  which  cannot  be 
derived  one  from  the  other. 

Thus,  a; +  2/ =  10, 

and  2a;  =  20  — 21/, 

are  not  independent  equations,  since  by  transposing  2y  in  the 
second  equation  and  dividing  it  by  2,  the  second  equation 
may  be  converted  into  the  first. 

But        Zx-2y-=b 

158.  Elimination  is  the  process  of  combining  two  equa- 
tions containing  two  unknown  quantities  so  as  to  form  a 
single  equation  with  only  one  unknown  quantity ;  or,  in 
general,  the  process  of  combining  several  simultaneous 
equations  so  as  to  form  equations  one  less  in  number  and 
containing  one  less  unknown  quantity. 


V  are  independent  equations. 


164  ALGEBRA. 

159.  Value  of  Simultaneous  Equations.  In  simulta- 
neous equations  we  have  given  the  relations  of  a  set  of 
unknown  quantities  to  known  quantities  in  the  shape  of  a 
group  of  equations.  By  the  use  of  the  methods  of  Chapters 
VI.  and  XI.,  and  by  elimination,  we  reduce  this  complex  set 
of  relations  to  more  and  more  simple  ones,  till  at  last  we 
arrive  at  relations  so  simple  that  the  value  of  each  unknown 
quantity  can  be  perceived  at  once. 

160.  Methods  of  Elimination.  There  are  three  principal 
methods  of  elimination : 

I.  Addition  and  Subtraction. 
II.  Substitution. 
III.  Comparison. 

These  methods  are  presented  to  best  advantage  in  conneo 
tion  with  illustrative  examples. 


Ex.  Solve  \    ^ 

{  9x  — 


161.    I.  Elimination  by  Addition  and  Subtraction. 

\2x  +  by  =  1b  , (1) 

42/  =  33 (2) 

Multiply  equation  (1)  by  4,  and  (2)  by  5, 

48a;  +  20y  =  300 (3) 

45a;  -  202/  =  165  . (4) 

Add  equations  (3)  and  (4)  93a;  =  465 

Divide  by  93,  a;  =  5,  Boot 

•    Substitute  for  x  its  value  5,  in  equation  (1), 
60  +  5^  =  75 
.  • .  2/  =  3,  Boot. 

Since  y  was  eliminated  by  adding  equations  (3)  and  (4), 
this  process  is  called  elimination  by  addition. 

The  same  example  might  have  been  solved  by  the  method 
of  subtraction. 


SIMULTANEOUS  EQUATIONS.  165 

Thus,  multiply  equation  (1)  by  3,  and  (2)  by  4, 

36a;  +  15^  =  225 (6) 


Subtract  (6)  from  (5), 


36a;  -  IQy  = 

132 

31y  = 

=  93, 

y  = 

=  3, 

and    X  = 

=  5. 

(6) 


It  is  important  to  select  in  all  cases  the  smallest  multipliers  that  will 
cause  one  of  the  unknown  quantities  to  have  the  same  coefficient  in  both 
equations.  Thus,  in  the  last  solution  given  above,  instead  of  multiplying 
equation  (1)  by  9,  and  (2)  by  12,  we  divide  these  multipliers  by  their  com- 
mon factor,  3,  and  get  the  smaller  multipliers,  3  and  4. 

Hence,  in  general, 

Multiply  the  given  equations  by  the  smallest  numbers  that  will 
cause  one  of  the  unknown  quantities  to  have  the  same  coefficient  in 
both  equations; 

If  the  equal  coefficients  have  the  same  sign,  subtract  the  equations; 
if  they  have  unlike  signs,  add-  the  equations. 


EXERCISE   55. 

Solve  by  addition  and  subtraction — 

1.  Sx-2y  =  l.  6.  Sx-2y  =  4. 

x  +  y  =  2.  5x  —  4y=--7, 

2.  2x-7y  =  9.  7.  2y  +  x  =  0. 

bx  -+-  Zy  =  2.  4x  +  Qy  =■■  —  3. 

S.ix  +  Sy  =  l.  8.     9x-Sy  =  6. 

2x-6y  =  S.  15x  +  12y  =  2. 

4.  5x-Sy  =  l.  9.  4x-Qy  +  l=0. 

Zx-\-5y  =  21.  5x-7y  +  l=0. 

6.    x  +  5y=-3.  10.  Sx  +  5y  =  Q. 

7x-^Sy  =  e.  &y+2x  =  ll. 


166  ALGEBRA. 

14.^  +  2/^1. 
11.  5ic-32/  =  36.         ,  ^       ^ 

7a;  — 52/ =  56.  3a;   ,  5?/      ^ 

2        8 

^_2/_  3^^2^^7 

4      9        *  4    '    5       2" 

13.^-^  =  3.  16.^^-^2/^-6. 

3      5  0        9 

^  +  ^  =  8.  5^_^  =  _6. 

5      2  4        6 

162.     II.  Elimination  by  Substitution. 

Ex.   Solve  5x  +  2«/  =  36 (1) 

2x-f  32/-43 (2) 

From  (1),  bx  =  36-2^ 

.-.       cr^^^-^^V (3) 

5 

In  eq.  (2)  substitute  for  a;  its  valno  given  in  (3), 
2(^«  -  ^11)  +  .V  =  43 

^2^^  +  33,  =  43 
6 

72  -  4?/  +  15^  =  215 

ll.V  =  143 

^  --=  13 

Substitute  for  y  in  (3),  a;  =  ^^  ~  ^-  =  2 

5 

Hence,  in  general, 

7n  one  0/  ^/le  ^iwn  equations   obtain  the  value  of  one  of  the 

unknown  quantities  in  terms  of  the  other  unknown  quantity ; 

Substitute  this  value  in  the  other  equation  and  solve. 

*  Clear  of  fractions  before  eliminating. 


SIMULTANEOUS  EQUATIONS.  167 


EXERCISE 

:  66. 

Solve  by  substitution. 

1.  3x-2/  =  2. 
2a;  +  52/  =  7. 

„.  ^-4^-n 

2.  3a;-42/  =  l. 
Ax-by  =  l. 

hi"- 

3.  2a:  +  32/  =  l. 

■''T-r^^ 

3a;  +  42/  =  2. 
4.    2/- 3a;  =9. 

5       10 
.,.1-1  =  3,. 

2a;  +  72/  =  —  6. 
5.  3a;  +  102/  =  -  L 

2x  +  72/  =  -i 

roi-*- 

6.  7a;  +  82/  - 19. 

6a;  +  62/  =  13i. 

7.  4a; +52/ =  10. 
7a;  +  32/  =  6. 

...  f -1^.,. 

8.  6a;-52/  =  3. 
5a;-62/  =  8. 

IS.  7.- 1?. 33. 

3 

9.  5a;- 1=42/. 
72/+19  =  3a;. 

6, +1^-9. 

i«..  5+1^10. 

...f=i.u 

5-^'«- 

32/      2j; 
8       3 

*  Clear  of  fractious  before  eliminating. 


168  ALGEBRA. 

163.  III.  Elimination  by  Comparison. 

Ex.   Solve  2a:-32/  =  23    .    .    .' (1) 

5x  +  2y  =  2d (2) 

From  (1)  2x  =  23  +  Si/ (3) 

From  (2)  6a;  =  29  -  2^ (4) 

From  (3)  x  =  23^^ (5) 

From  (4)  x  =  29_=J^ (6) 

5 

Equate  the  two  values  of  x  in  (5)  and  (6), 

23  +  3.y  _  29  -  2.v 
2  5      ' 

Hence,  115  +15^-58-4^ 

1^  =  -  57 
y  =  —  3,  Boot. 

23  —  9 
Substitute  for  y  in  (5),  x  =  — - —  =  7    Boot. 

Hence,  in  general, 

Select  one  unknown  quantity,  and  find  its  value  in  terms  of  the 
other  in  each  of  the  given  equations  ; 

Equate  these  two  values^  and  solve  the  resulting  equation. 

EXERCISE   57. 

Solve  by  comparison — 

1.  bx  —  Zy  =  2.  5.  3a; +  2^/  =  5. 

x  +  2y  =  Z,  4a;-32/  =  -l|. 

2.  y-2x  =  Z.  6.  Zx-by  =  ld. 
Zy  +  x  =  2.  4a: +  72/ =  —2. 

3.  x-2y  =  &.  7.  ^x-V\y  =  5. 
5x  +  32/  =  4.  ^x  +  \y  =  b. 

4.  2a;  +  7t/  +  l-0.  8.  \x-\y  =  l, 
6a;  +  82/-7  =  0.  i^-i2/  =  l. 


9. 

!-!==- 

M-- 

10. 

^^y-^^. 

?-«*• 

11. 

3       ^      12 

^-2y  =  -|. 

SIMULTANEOUS  EQUATIONS.  169 


2  3 

=c  +  3      2y  , 

4  6"     *• 

3  4 

2y^^+2j,^^^ 
o  2t 

14.  ^±^-L 


2/-3      6 

2a; +  5 


y  +  x 
164.  Fractional  Simultaneous  Equations. 


=  -9. 


Ex.  Solve 


2x  +  y  ^b  2y  +  x 

4'     ~4  4 

2x  —  y_  2^/  — a; 

—J— -2/  ^ 


Transpose  terms,  J 


Clear  the  given  equations  of  fractions, 

{4a:  —  2a:  —   y  =  5  —  2^  —  a; 
12  -  6a;  +  3y  =  '[2y  -  8^/  +  4a; 
3a:  +  2/  =  5 
-  10a:  -  2/  =  -  12 
-  7a:  =  -  7 
a;  =  l 

y  =  2. 
Hence, 

Simplify  each  of  the  given  equations  by  the  methods  used  (Chap' 
ter  XL)  for  an  equation  of  one  unknown  quantity  ; 

Solve  the  resulting  equations  by  one  of  the  three  methods  of  elimr 
ination. 


170  ALGEBRA, 


EXERCISE   58. 

Solve— 


3  4  •  '  bx-Zy  +  Z       5* 

3a; +  1      y^Q  2a;-y  ,    x  —  2y 

4  3-  3       "^       4 


3/-2  _  .  ^^-^2/ 


=  1. 


2.    a;-^-^=5. 
7 


12 


4   _  ^  +  1Q  ^3  7.  0.7a;  -  0.02.V  =  2. 

^  0.7a; +  0.022/ =  2.2. 

3    4a;  +  53/_^_  8.  0.4a; -  0.31/ - 0.7. 

40 

0.7a; +  0.22/ =  0.5. 
2a;  — V 

— ^  +  2y  =  f  ^       2a;  +  1.57/  - 10. 

o       -,       .      o  0.3a; -0.052/ =  0.4. 

5              7  10.  0.5a; +  4.52/ =  2.6. 

%  +  a;  1.3a;^  3.12/ =  1.6. 

11     "^^~^* 

11.  0.8a; -0.72/ =  .005. 

5.?^^  +  5^^  =  2:  2.  =  32/. 
5               11 

ii±lZli_25_,  ='2.       3:r  =  0.42/ +  0.1. 

4              3  ""^-  .06y  =  0.5a;-.02. 


18.        ^-^^       =^  +  13. 
,    2-3a: 

102/  +  1  ^ 2a; +  3 

5       "'       .+3-i±^ 


SIMULTANEOUS  EQUATIONS.  171 

x  +  y     5 

J 2__JL2 3_^ 

Hi  If 

16.  (x--5)(3/  +  3)-(a;-l)(2/  +  2).: 
an/  +  2x  =  a;(2/  +  10)  +  72i/. 

5  3               4 

2y  +  6  4a;-hy  +  6 

3  8 

j7^  62/4-5  3a;  +  5j-  _92/-4 


8  6x  -  22/  12 

'  +  3         a;  +  2/    _  ^y  + 
4  3x-2y~      8 


18    3a;-2^6a;-5      a;  +  2/  +  6i 
5  10  6a;  +  2/     ' 

Sy-2^2y-5         S  +  7x 
12  8  103/-3a;' 

165.  Literal  Equations. 

Ex.   Solve  ax  +  hy  =  c (1) 

o!x^h'y^d (2) 

Multiply  (1)  by  a\  and  (2)  by  a, 

aa'x  +  a'hy  =  a'e (3) 

aa^x  +  ah'y  =  a& (4) 

Subtract  (4)  from  (3),      {a'h  —  ab')y  =  a'c  —  ac^ 

•    y  =  «^c  —  a(/ 
'  '  ^     a'b-  ab^ 
Again,  multiply  (1)  by  6^,  (2)  by  6, 

ab^x  -f  66'y  =  b^G (5) 

a^bx  +  bb^y  =  bc^ (6) 

Subtract  (6)  from  (5),      {ab'  -  a'b)x  =  b'c  -  b& 

.  b'c-bc\ 

ab'-a'b 


172  ALGEBRA. 

It  is  to  be  observed  that  after  finding  the  value  of  a:,  it  is 
best  not  to  find  the  value  of  y^  as  in  numerical  equations,  by 
substituting  the  value  of  x  in  one  of  the  original  equations 
and  reducing,  but  rather  by  taking  both  the  original  equations 
and  eliminating  anew. 

EXERCISE   59. 

Solve— 

1.  3a:  +  42/  =  2a.  11.  (a  +  1>— %  =  «  +  2. 
5a;  +  62/  =  4a.  (a  —  V)x  +  Zhy  =  9a. 

2.  2aa;  +  Zhy  =  4a6.  12.  (a  +  6)a;  +  ci/  =  1. 
bax  +  462/  =  Sa6.  ex  +  (a  +  6)2/  =  1. 

3.  ax-\-hy  =  1.  I3.  _^ [.  _1_  =  2. 

a  +  6       a  — 6 


a'a;  +  6'2/  =  1. 

4.  x  —  y  =  2n. 
mx--ny  =  m' 


14.  (a  —  b)x  =  (a  —  d)y. 
x  —  y  =  l. 


5.  2bx -]- ay  =  4b  +  a.  .        la     ,  /     ,   i.n 

^  ^    (a  —  o)a;  +  (a  +  o)v      ^ 
a6a;  -  laby  =  46  +  a.  15.  ^ ~^:^^ ^  ■^* 

6.  ax~by  =  a^-\-b\  ax  -  2by  =  a^  -  2b\ 


bx  +  ay  =  2(a'^  +  6'). 
T.  ax  +  by  =  c.  a  +  6    '    a— 6 


16.  ^z:A  +  lziA=._i. 


mx  +  ny^d.  a;  +  2a     y-2b^a'-hb' 

8.  bx  +  ay  =  a-^b.  a  —  b       a  +  b       a'  —  b^ 
ab(x-y)='a'-b\  ^^    rp-l       y-a  ^^ 

9.  c^x  —  d'y^c  —  d.  *   6 - 1       6  —  a 
C£Z(2(^a;-C2/)=2d^-c'.  x-\-l    ^    y-1  ^1 
a;-hm^  n  ^  l-«      ^ 

*  2/-n      m'  18.  (ic-l)(a+6)=a(2/+a+l> 
jc  +  2/  =  2n.  (2/+l)(a-6)=6(a;-6-l). 


SIMULTANEOUS  EQUATIONS.  173 

166.  Three  or  More  Simultaneous  Equations.  If  three 
simultaneous  equations,  containing  three  unknown  quanti- 
ties, be  given,  we  may  take  any  pair  of  the  given  equations 
and  eliminate  one  of  the  unknown  quantities;  then  take  a 
different  pair  of  the  given  equations,  and  eliminate  the  same 
unknown  quantity.  The  result  will  be  two  equations  with 
two  unknown  quantities,  which  may  be  solved  by  the  meth- 
ods already  given. 

In  like  manner,  if  we  have  n  simultaneous  equations,  con- 
taining n  unknown  quantities,  by  taking  different  pairs  of 
the  n  equations,  we  may  eliminate  one  of  the  unknown  quan- 
tities, leaving  n  —  1  equations,  with  n  —  1  unknown  quantities, 
and  so  on. 

'3a;  +  42/-5z=32 (1) 

Ex.  Solve  ■  4a; -52/ +  32=18 (2) 

bx-Zy-4.z  =  2 (3) 

If  we  choose  to  eliminate  z  first, -multiply  (1)  by  3,  and  (2)  by  5, 

9a;  +  12y  -  150  =  96 (4) 

20a;  -  25^  +  150  =  90 (6) 

Add  (4)  and  (5),  29a;  -  13^  =  186 (6) 

Also  multiply  (2)  by  4,  (3)  by  3, 

16a;  -  20y  +  120  =  72 (7) 

15a;  -  9y  -12z  =  6 (8) 

Add  (7)  and  (8),  31a;  -  29y  =  78 (9) 

We  now  have  the  pair  of  simultaneous  equations, 
r29a;-132/  =  186 
l31a;-29^  =  78 
Solving  these,  obtain  a;  =  10 

Substitute  for  :p  and  y  in  equation  (1), 

30  +  32  -  53  =  32, 
0  =  6. 


174  ALGEBRA. 

EXERCISE  60. 

Solve — 

1.  x  +  y  +  z-=6.  9.  ix+.iy  +  lz  =  2, 
^x  +  2y  +  z  =  10.  ix  +  iyi-  |z  =  9. 
3x  +  2/  +  3z  =  14.  ia;  +^2/  +  ^2  =  ^^ 

2.  3x-2/-2z  =  ll.  10.  2x  +  2y-z  =  2a. 
4x-2y  +  z  =  -2.  Sx-y-z  =  Ab. 
6a:-2/  +  3z  =  -3.  5a;  +  3^/ -  3z  -  2(a  +  6), 

3.  5a;-6i/  +  2z-5.  2x      3^  _  4z  _  ^g^ 
8x  +  42/-5z  =  5.  '345 

9a;  +  52/  -  6z  =  5.  ^  _  §/  -f  ??-  =  —  5. 

4.  3x  +  22/-z  =  9.  6        8        4 

7.  +  52/  +  2z  =  3.  ?^_Z^  +  ^  =  _41. 


2a;-72/  +  5z  =  0. 


2        5       10 
12.  x  +  y-{-2z  =  2(a  +  6). 


5.  2x  +  3y  =  7. 
Sy  +  4z  =  9.  x  +  z  +  2y  =  2(a  +  c). 
6x  +  6z  =  W.  y  +  z  +  2x  =  2(b+c). 

6.  2x  +  4y  +  Bz  =  e,  13.  a;  +  ^-z  =  3-a-6. 
62/  -  3x  +  2z  =  7.  a;  +  z  -  2/  =  3a  -  6  - 1. 
3a;-82/-7z  =  6.  1/ 4-z-a;  =  36-a-l. 

T.  a;4-32/  +  3z  =  l.  14.  3x  +  22/  =  ya. 

32;_5z  =  l.  6z-2a;  =  |6. 

92/ +  lOz  +  3a;  =  1.  52/ -  13z  +  a;  =  0. 

8.  w  +  v-w  =  4.  15.  -x  +  y  +  z  +  v  =  a. 

u  +  v-x  =  l.  x-y-^z  +  v  =  b. 

v-{-w  +  x  =  S.  x-\ry-'z-{-v  =  c. 

^_-^  +  a;  =  5.  x-^y  +  z~v  =  d. 


SIMULTANEOUS  EQUATIONS. 


176 


167.  Use  of  —  and  —  as  Unknown  Quantities. 
ac  y 

Some  equations  which   would  otherwise  be  difficult  of 

solution  are  readily  solved  by  regarding  -  and  -  as  the  un- 

X  y 

known  quantities  and  eliminating  them  by  the  usual  methods. 


Ex.  1   Solve 


^5  +  i^  =  49 
X      y 

^  +  -  =  23. 

X      y 


(1) 
(2) 


Multiply  (1)  by  7,  and  (2)  by  5, 


Subtract  (4)  from  (3), 


^  +  ^  =  343 
X       y 


35  ^  15 
X      'y 

76 

y 

,       1 


115 


=  3 

y 

y  =  i,  Root, 
Substitute  the  value  of  ^  in  (2),  hence, 

X  =  If  Boot. 


(3) 
(4) 


Ex.  2.  Solve 


2x      32/ 

?_!-  =  ?? 
4y       4 


X 


(1) 

(2) 


When  X  and  y  in  the  denominators  have  coefficients,  as  in 
this  example,  it  is  usually  best  first  to  remove  these  coeffi- 
cients by  multiplying  each  equation  by  the  L.  C.  M.  of  the 


176 


ALGEBRA. 


coefficients  of  x  and  y  in  the  denominators  of  that  equation. 
Hence, 

Multiply  (1)  by  6,  and  (2)  by  4, 


X 

.10^ 

y  ' 

=  66 

8 

.     X 

_  1^  __ 
y 

=  29 

Multiply  (4)  by  10, 
Add  (3)  and  (5), 

80  _ 

10 

-  — -  = 

=  290  .   .   .   

X 

y 

89 

—  = 

X 

=  356 

, 

'  .X  = 

=  h 

from  (4), 

• 

y  = 

-h 

EXERCISE  61. 

Dive — 

i.i+?=i. 

X      y 

X      y 

?+5=i. 

X    y 

— -?-3 
X       y 

2.^  +  ^  =  3. 

X      y 

'■l-Ay^'- 

5  +  5  =  2. 

X      y 

-2+l  =  -5. 
3x      2y 

8.1-2=7. 

X      y 

'■i-t''^- 

?  +  ^=i. 

X      y 

i-l-''^- 

4.5  +  6^2. 
X     y 

8.i  +  Ul. 

X      y      n 

X     y 

1      1 

=  n. 

»     y 

(3) 
(4) 
(5) 


SIMULTANEOUS  EQUATIONS,  177 


X      y          a 

15. 

^    y 

b      a  _  b  —  a 
X      y          a 

2/      z 

lO.  \-  -  =  m^  4-  n. 

X       y 

X      y 

16. 

Yx      ¥y^Vz~^- 

11.  f  +  A=2. 
bx      ay 

cc      2/      zz 

6      g  ^  g^  4-  6\  A_A_1:^15 

X      y  ab  2x      4y      3z 


^L±l-i.^-^  ^0^  17    1_1_1  =  1 


12.  — ^ 1 =  2a. 

X  y 


13.  52/ —  3a;  =  7a;2/. 


X      y      z      a 


(^  ,   b  ,  ,  1111 

^      y  y      z      X      h 


1111 


15a:  +  602/ =  16x2r.  z      x      y      c 

14.^  +  ^4-^  =  2.  18.^  +  ^-^^Z. 

X      y      z  X      y    .z 

2      1,1^  g  ,  c       6 

4--=7.  -4- =m. 

x      y      z  X      z       y 

3,2,5..  6, eg 

-4 h-  =  14.  -4 =n. 

X      y      z  y      z      X 

19.  52/z  4-  6a:z  —  3x2/  =  ^xyz. 

Ayz  —  dxz  -\-  xy=^  19xyz. 

yz  —  12xz  —  2xy  =  dxyz, 
12 


CHAPTER   XIII. 

PROBLEMS    INVOLVING   TWO   OR    MORE    UN- 
KNOWN   QUANTITIES. 

168.  In  the  solution  of  problems  we  are  sometimes  obliged 
to  employ  more  than  one  letter  to  represent  the  unknown 
quantities.  We  must  always  obtain  from  the  conditions  of 
the  problem  as  many  independent  equations  as  there  are 
letters  thus  involved. 

Ex.  1.  Find  two  numbers  such  that  the  greater  exceeds 
twice  the  less  by  1,  and  twice  the  greater  added  to  three  times 
the  less  equals  23. 

Let  X  =  the  greater  number, 

and  y  =  the  less. 

Then,  x-'ly  =  \    \ 

1  o     ,  o  ,  ^  OQ  I  f^®°^  ^^  conditions  of  the  problem. 

Solving  these  equations  by  any  of  the  common  methods  of  elimination, 
one  obtains  a;  =  7  ;  ^  =  3. 

Hence  7  and  3  are  the  numbers  required. 

Ex.  2.  There  is  a  fraction  such  that  if  2  be  added  to  both 
numerator  and  denominator,  it  becomes  \ ;  but  if  7  be  added 
to  both  numerator  and  denominator,  it  reduces  to  |.     Find  it. 


Let  —  represent  the  fraction. 

y 

Then,                                      ^  +  2 
2/  +  2 

1 
2' 

and                                         ^  +  7_ 
2/  +  7 

2 
3* 

Clearing  these  equations,  and  collecting  like  terms. 

2x-y  = 

-2 

Zx-1y  = 

-7 

The  solution  shows  x  =  Z  and  ^  =  8. 

Therefore  |  is  the  required  fraction. 

178 

PROBLEMS.  179 

Ex.  3.  Two-fifths  of  A's  age  is  3  years  less  than  |  of  B's 
age ;  but  |  of  A's  age  equals  B's  age  10  years  ago. 

Let  X  =  the  No.  of  years  in  A's  age. 

y  =  the  No.  of  years  in  B's  age. 
and  y  —  10  =  B's  age  ten  years  ago. 

Then,  |a:  =  |y  -  3, 

^x  =  y  -  10. 
Ciearing  and  solving,         a;  =  45  and  y  =  35. 
Thtts,  A  is  45,  and  B,  35  years  of  age. 

EXERCISE  62. 

1.  Find  two  numbers  whose  sum  is  23  and  whose  difference 
is  5. 

2.  Twice  the  difference  of  two  numbers  is  6,  and  \  their 
sum  is  Z\.     What  are  the  numbers? 

3.  Find  two  numbers  such  that  twice  the  greater  exceeds  5 
times  the  less  by  6  ;  but  the'  sum  of  the  greater  and  twice  the 
less  is  12. 

4.  If  1  be  added  to  the  numerator  of  a  certain  fraction,  its 
value  becomes  \ ;  but  if  1  be  subtracted  from  its  denominator, 
its  value  is  |.     Find  the  fraction. 

5.  Two  pounds  of  flour  and  five  pounds  of  sugar  cost  31 
cents,  and  five  pounds  of  flour  and  three  pounds  of  sugar 
<Jost  30  cents.     Find  the  value  of  a  pound  of  each. 

6.  What  is  that  fraction  whose  numerator  is  3  less  than  the 
denominator,  but  5  times  the  numerator,  less  1,  is  equal  to  3 
times  the  denominator? 

7.  A  number  consists  of  two  digits  whose  sum  is  13,  and  if 
4  is  subtracted  from  double  the  number,  the  order  of  the 
digits  is  reversed.     Find  the  number. 

8.  A  man  hired  4  me«i  and  3  boys  for  a  day  for  $18;  and, 
for  another  day,  at  the  same  rate,  3  men  and  4  boys  for  $17. 
How  much  did  he  pay  each  man  and  each  boy  a  day  ? 

9.  There  is  a  fraction  such  that  if  4  be  added  to  its 


180  ALGEBRA. 

numerator  it  will  become  i,  and  if  3  be  subtracted  from  the 
denominator  it  will  become  |.     What  is  the  fraction  ? 

10.  In  an  orchard  of  100  trees  there  are  5  more  apple  trees 
than  f  of  the  number  of  pear  trees.  How  many  are  there  of 
each? 

11.  A  farmer  sells  to  one  man  4  sheep  and  9  calves  for  $79. 
and  to  another,  at  the  same  rate,  7  sheep  and  5  calves  for  $63. 
Required  the  price  of  a  sheep  and  of  a  calf. 

12.  Three  times  A's  money  is  $60  more  than  4  times  B's ; 
and  \  of  A's  is  $20  less  than  ^  of  B's.     How  much  has  each? 

13.  One-seventh  of  A's  age  is  2  years  less  than  half  of  B's ; 
and  3  times  B's  age  is  equal  to  what  A's  age  was  1  year  ago. 
Find  their  ages. 

14.  Find  two  numbers  such  that  3  times  the  difference  of 
their  halves  is  4  more  than  4  times  the  difference  of  their 
thirds,  and  \  of  the  larger  is  equal  to  -J-  of  the  smaller. 

15.  In  8  hours  A  walks  8  miles  more  than  B  does  in  7 
hours;  and  in  12  hours  B  walks  3  miles  more  than  A  does 
in  10  hours.     How  many  miles  does  each  walk  in  one  hour? 

16.  A  party  of  boys  purchased  a  boat,  and  upon  payment 
discovered  that  if  they  had  numbered  3  more,  they  would 
have  paid  a  dollar  apiece  less ;  but  if  they  had  been  2  less, 
they  would  have  paid  a  dollar  apiece  more.  How  many  boys 
were  there,  and  what  did  the  boat  cost  ? 

Hint.  Let  x  =  the  number  of  boys,  and  ?/  the  number  of  dollars  each 
paid.    Then  xi/  represents  the  number  of  dollars  the  boat  actually  cost. 

17.  If  a  rectangle  were  3  inches  longer  and  an  inch  nar^ 
rower  it  would  contain  7  square  inches  more  than  it  now 
does ;  but  if  it  were  2  inches  shorter  and  2  inches  wider  its 
area  would  remain  unchanged.     What  are  its  dimensions? 

18.  Two  persons,  A  and  B,  can  perform  a  piece  of  work  in 
16  days.  They  work  together  for  four  days,  when  B  is  left 
alone  and  completes  the  task  in  36  days.  In  what  time 
could  each  do  it  separately? 

19.  If  A  gives  B  $10,  A  will  have  half  as  much  as  B ;  but 


PROBLEMS,  181 

if  B  gives  A  $30.  B  will  have  |  as  much  as  A.     How  much 
has  each  ? 

20.  A  train  maintained  a  uniform  rate  for  a  certain  dis- 
tance. If  this  rate  had  been  8  miles  more  each  hour,  the 
time  occupied  would  have  been  two  hours  less;  but  if  the 
rate  had  been  10  miles  an  hour  less,  the  time  would  have 
been  4  hours  more.     Required  the  distance. 

21.  A  certain  fraction  becomes  ^  if  If  be  added  to  both 
numerator  and  denominator;  but  ^  if  2|  be  subtracted  from 
numerator  and  denominator.     What  is  the  fraction  ? 

22.  If  the  greater  of  two  numbers  is  divided  by  the  less, 
the  quotient  is  3  and  the  remainder  3,  but  if  3  times  the 
greater  be  divided  by  4  times  the  less,  the  quotient  is  2  and 
the  remainder  20.     Required  the  numbers. 

23.  Find  two  fractions,  with  numerators  11  and  7  respec- 
tively, such  that  their  sum  is  3||-,  but  when  their  denomina- 
tors are  interchanged,  their  sum  becomes  3^^. 

24.  If  f  be  added  to  the  mimerator  of  a  certain  fraction,  its 
value  is  increased  by  ^ ;  but  if  2^  be  taken  from  its  denomi- 
nator, the  fraction  becomes  f.     Find  the  fraction. 

25.  The  sum  of  the  digits  of  a  certain  number  of  two 
figures  is  5,  and  if  3  times  the  units'  digit  is  added  to  the 
number,  the  order  of  the  (iigits  will  be  reversed.  What  is 
the  number? 

26.  There  are  two  numbers  consisting  of  the  same  two 
digits ;  the  difference  of  the  digits  is  •  1  and  the  sum  of  the 
numbers  is  121.     What  are  the  numbers? 

27.  The  sum  of  the  digits  of  a  number  of  two  figures  is  \ 
of  the  number,  but  if  18  be  added  to  the  number,  the  order 
of  the  digits  is  reversed.     What  is  the  number? 

28.  Twice  the  units'  digit  of  a  certain  number  is  2  greater 
than  the  tens'  digit ;  and  the  number  is  4  more  than  6  times 
the  sum  of  its  digits.     Find  the  number. 

29.  Of  a  number  consisting  of.  3  digits  the  tens'  digit  is  5, 
and  the  units'  digit  1  less  than   twice  the   hundreds'  digit, 


182  ALGEBRA. 

and  the  number  will  be  increased  by  99  if  the  order  of  the 
figures  be  reversed.     What  is  the  number? 

80.  What  is  that  number  of  three  figures,  the  first  and  last 
of  which  are  alike,  the  tens'  digit  is  1  more  than  twice  the 
sum  of  the  other  two,  and  if  the  number  is  divided  by 
the  sum  of  its  digits,  the  quotient  is  21  and  the  remain- 
der 4? 

31.  One  woman  buys  4  yards  of  silk  and  7  of  satin,  and 
another  at  the  same  rate  buys  5  yards  of  silk  and  5|-  of  satin, 
each  paying  $17.70.     What  is  the  price  of  a  yard  of  each  ? 

32.  Find  three  numbers  such  that  if  each  be  added  to  half 
the  sum  of  the  other  two,  the  three  sums  thus  resulting  will 
be  38,  40  and  42. 

33.  One  cask  contains  18  gallons  of  wine  and  12  gallons  of 
water;  another,  4  gallons  of  wine  and  12  of  water.  How 
many  gallons  must  be  taken  from  each  cask  so  that  when 
mixed  there  may  be  21  gallons,  half  wine  and  half  water? 

34.  A  gentleman  gave  a  sum  of  money  to  his  four  sons; 
giving  to  the  eldest  half  the  sum  of  the  shares  of  the  other 
three;  to  the  second,  one  third  the  sum  of  the  other  three 
shares;  and  to  the  third,  one  fourth  the  sum  of  the  other 
three  shares.  The  share  of  the  eldest  exceeded  that  of  the 
youngest  by  $70.  What  was  the  whole  sum,  and  what  did 
the  eldest  receive  ? 

35.  A's  money  with  twice  B's  and  2|-  times  C's  is  $340;  f 
of  A's  and  B's  together  is  equal  to  $4  less  than  ^  of  C's,  and 
A  and  C  have  ^  of  B's  and  C's  together.  How  much  has 
each? 

36.  A  merchant  found  that  he  had  $37  in  silver  dollars, 
half-dollars  and  quarters;  altogether  he  had  84  pieces.  He 
also  noticed  that  J  of  the  dollars,  -J-  of  the  halves,  and  J  of 
the  quarters  would  have  been  worth  only  $7.50.  How  many 
of  each  did  he  have  ? 

37.  There  are  two  fractions  having  the  same  denominator, 
whose  sum  is  f .     If  1  be  added  to  each  numerator,  the  sum 


PROBLEMS.  183 

of  the  fractions  becomes  IJ,  but  if  1  be  subtracted  from  each 
numerator,  their  difference  is  ^.     Find  the  fractions. 

Hint.     Let  -  be  the  larger,  and  ^  the  less  fraction. 
z  z 

38.  A  gives  to  B  and  C  as  much  as  each  of  them  has ;  after- 
ward, B  gives  to  A  and  C  as  much  as  each  of  them  now  has; 
and  then  C  gives  to  A  and  B  as  much  as  each  of  them  has, 
when,  finally,  they  each  have  $16.   How  much  had  each  at  first? 

39.  Three  men.  A,  B,  and  C,  have  each  a  certain  sum,  such 
that  if  A  should  give  B  $40,  he  would  have  3  times  as  much 
as  A  had  left;  if  B  should  give  C  $10,  he  would  have  2^ 
times  as  much  as  B  had  left ;  and  if  C  should  give  A  $30, 
he  would  have  If  times  as  much  as  C  had  left.  How  much 
had  each  ? 

40.  To  distribute  $50  to  some  men,  women,  and  children,  I 
gave  $2  to  each  man,  $2.50  to  each  woman,  and  $1.50  to  each 
child,  with  $4  left  over ;  if  I  had  given  $2.50  to  each  man  and 
woman,  and  $1.20  to  each  child,  I  would  have  had  $3  left; 
but  when  I  gave  $1.75  to  each  man,  and  $2.25  to  each  woman 
and  child,  nothing  remained.    How  many  of  each  were  there  ? 

41.  A  boy  spent  $3  for  oranges,  some  at  30  cents  a  dozen, 
and  some  at  40  cents ;  he  sold  them  at  3  cents  apiece,  and 
gained  24  cents.    How  many  dozen  of  each  kind  did  he  buy? 

42.  A  rectangle  has  the  same  area  as  another  4  yards  longer 
and  2  yards  narrower,  and  the  same  as  a  third  3  yards  shorter 
and  2|  yards  wider.     What  are  its  dimensions  ? 

43.  If  a  rectangle  were  made  3  feet  shorter  and  1^  feet 
wider,  or  if  it  were  7  feet  shorter  and  5\  feet  wider,  its  area 
would  remain  unchanged.     What  are  its  dimensions? 

44.  The  fore  wheel  of  a  carriage  makes  6  more  revolutions 
than  the  hind  wheel  in  going  180  yards,  but  if  the  circumfer- 
ence of  the  fore  wheel  were  increased  by  i  of  itself,  and  that 
of  the  hind  wheel  decreased  by  ^  of  itself,  the  fore  wheel 
would  make  6  revolutions  less  than  the  hind  wheel  in  the 
same  distance.     Find  the  circumference  of  each. 


184  ALGEBRA. 

45.  A  boy  rows  18  miles  down  a  stream  and  back  in  12 
hours:  he  finds  that  he  can  row  8  miles  with  the  stream 
while  he  rows  1  mile  against  it.  Find  his  rate  and  that 
of  the  stream. 

Solution.     Let  x  =  the  number  of  miles  the  boy  can  row  an  hour  in 

still  water. 
And  2/  =  tli6  number  of  miles  the  stream  flows  an  hour. 

Then  a;  +  2/  =  his  rate  down  stream, 

and  x—y=       "       up  '' 

=  number  of  hours  required  to  row  down. 


up. 


12,         or     -^—  +  — ^—  =  2 (1) 

a;  +  ^       X  -  y  x  -\-  y       x  —  y 

He  rows  down  three  times  as  fast  as  he  does  up  ;  hence,  he  requires  only 
\  as  much  time. 

-       18_  _  1/     18     \         __         3  1 

+  y       Z\x-y) 


18 


x-^  y 

18 

x-y 

Hence, 

18 

1      18 

.    .    .    .(2) 


X -^  y       6\x  —  y  J  x^y       x  — 

Solving  (1)  and  (2)  by  method  of  exercise  61, 

a;  +  ?/  =  6,     and  x  —  y  =-  2.    Hence  a;  =  4,     and  y  =  % 
Thus  he  rows  4  and  the  stream  flows  2  miles  an  hour. 

46.  A  boatman  rows  20  miles  down  a  river  and  back  in  8 
hours :  he  can  row  5  miles  down  the  river  while  he  rows  3 
miles  up  the  river.     Find  his  rate  down  stream. 

47.  A  man  rows  for  4  hours  down  a  stream  which  runs  at 
the  rate  of  3  miles  an  hour :  in  returning  it  takes  14|  hours 
to  reach  a  point  3  miles  below  his  place  of  starting.  Find 
the  distance  he  rowed  down  the  stream  and  his  rate  in  still 
water. 

48.  A  man  rows  down  a  stream  20  miles  in  2|  hours,  and 
rows  back  only  f  as  fast.  At  what  rate  does  the  water 
flow? 


PROBLEMS.  185 

49.  Two  bins  contain  a  mixture  of  corn  and  oats,  the  one 
twice  as  much  corn  as  oats,  and  the  other  3  times  as  much 
oats  as  corn.  How  much  must  be  taken  from  each  bin  to  fill 
a  third  bin  holding  40  bushels,  half  to  be  oats  and  half  corn  ? 

50.  A  and  B  are  walking  along  2  roads  which  cross  each 
other :  when  A  is  at  the  point  of  crossing  B  has  560  yards 
yet  to  walk  before  reaching  it ;  in  4  minutes  they  are  equally 
distant  from  this  point,  and  again  in  24  minutes  more  they 
are  equally  distant  from  it.     How  fast  does  each  walk? 

51.  A  sets  out  from  P  toward  Q,  and  3  hours  later  B  starts 
from  Q  toward  P,  traveling  2  miles  an  hour  faster  than  A. 
When  they  meet  B  has  walked  3  miles  more  than  A.  If  A 
had  walked  a  mile  an  hour  faster,  they  would  each  have 
walked  an  hour  less  than  they  did,  and  B,  7  miles  less  than 
A.     How  fast  and  how  far  did  each  walk? 

52.  A,  B,  and  C  were  engaged  to  mow  a  field.  The  first 
day  A  worked  3  hours,  B,  2  hours,  and  C,  4,  and  together  they 
mowed  an  acre ;  the  second  day  A  worked  6  hours,  B,  13,  and 
C,  2,  and  they  mowed  2  acres  ;  the  third  day  they  worked  12, 
8,  and  8  hours  respectively,  and  mowed  3  acres.  How  many 
hours  would  each  alone  have  required  to  mow  an  acre? 

53.  A  and  B  run  a  mile  race.  In  the  first  heat  A  gives  B  a 
start  of  60  yards  and  wins  by  50  yards.  In  the  second,  A 
gives  B  a  start  of  25  seconds  and  is  beaten  by  3  seconds. 
Find  the  rate  of  running  of  each. 

Solution.     Let  x  =  the  number  of  yards  A  runs  in  1  sec. 
y  ^      ii  u        '*      B        **       " 

r  B  runs  1650  yards  while  A  runs  1760. 
Ist  ^  .      1660  _  1760.  ' 

r  B  runs  1760  -  25?/  yards  while  A  runs  1760  -  3a;. 
2d  j     ^      1760  -25.y  _^  1760  -  3a; 

or,  lM_lI6q=22.    Solving,  a;  =  5i;y  =  fifr 

^  a; 

That  is,  A  runs  5|,  and  B,  5  yards,  a  second. 


186  ALGEBRA. 

54.  In  a  mile  race  A  gives  B  a  start  of  40  yards  and  wins 
by  70  yards.  In  another,  he  gives  B  a  start  of  32  seconds, 
and  is  beaten  by  5J  seconds.     Find  the  rate  of  each. 

55.  A  and  E  run  a  mile.  In  the  first  race  A  receives  48 
yards  the  start,  and  is  beaten  by  1  second ;  in  the  next  race, 
A  receives  12  seconds  start,  and  wins  by  11  yards.  How 
many  minutes  does  each  require  to  run  a  mile? 

56.  In  a  thousand-yard  race  A  gives  B  a  start  of  5  seconds, 
and  wins  by  31J  yards.  But  if  he  gives  B  a  start  of  100 
yards,  B  wins  by  6  seconds.  How  long  would  it  take  each 
to  run  a  mile? 

57.  A  in  a  given  time  accomplishes  |  as  much  as  B  in  the 
same  time,  and  they  together  engage  to  reap  a  field  in  15  days. 
Before  the  work  is  completed  they  are  obliged  to  call  in  C, 
who  could  have  done  the  entire  task  in  36  days.  But  if  A 
had  begun  4  days  earlier  and  B  a  day  later,  and  if  they  had 
called  in  C  two  days  later  than  they  did,  they  could  have 
finished  the  field  in  the  promised  time.  How  many  days  did 
C  help,  and  how  many  days  would  it  have  taken  A  alone  to 
reap  the  field  ? 

58.  A  gives  B  and  C  each  half  as  much  as  each  already 
has ;  then  B  gives  A  and  C  each  half  as  much  as  each  now 
has ;  afterward,  C  gives  A  and  B  half  as  much  as  each  has, 
when  they  have  $27  apiece.     How  much  had  each  at  first  ? 

EXERCISE   63. 

REVIEW. 
'     1.  Tell  the  degree  of  each  term  of — 

bxi^  —  4x^2/"^  —  11a:  —  xy  +  xy^  -  T^y  +  Zx^  —  7y  +  11. 
2.  Find  the  H.  C.  F.  and  L.  C.  M.  of— 

4a;3  -  13a;  +  6  and  ^x?  -  2x^  -  9. 


Simplify — 


3  4a:  —  5  _  4  +  X    .  2  _  a;  —  5  . 
'45  30         3  18 


REVIEW.  187 


M    1  -5x   ^  3a;  -h  5  ^  2a;  -  3 

*  6a;'  -  6       4a;  +  4      3  -  3a; 


5.-2_^        1 


(a;  -  ly      {l-xy      1-x      X 


6     2a;'  -  a;'  -  a;  -  3  ^ 

'  2a;3  -  5a;2  +  a;  +  3  ' 


7.  if(f^'-¥^-l). 


9.  1 


i(ta;»  -  fa;  -  2)  ^  _  _a_ 

a  +  a; 


Solve— 


10.  1|  +  a;  =  ^^  +  -  -  ^-=^- 
^  2  3  5 

3       2V        2/       3^3       \  2    // 


3a;  -  2      5a;  +  14       3  -  2a; 


^^-       6 

7a;  +  15 

4           -      ^^• 

13.      ^      - 
a;-2 

a;  +  1      X- 
x-1       X- 

-8      a;-9 
-6      x-7 

14.  3^-i^  =  i. 
2        3       9 

15.  2y  -  a;  =  4a^. 

f +  f  =3i. 

16  a^-2      10-a;^.v-10. 
•      5  3  4 

2.V  -f  4  _  2a;  -f-  y  ^  x  +  13. 
3  8  4      ' 


17.  (a  -  h)x  +  (a  +  6)^/  =-  a  +  6. 
(a:  -  y)  (a*  -  6')  =  a'  +  6^ 


CHAPTER    XIV. 
INEQUALITIES. 

169.  An  Inequality  is  a  statement  in  symbols  that  one 
algebraic  expression  represents  a  greater  or  less  number  than 
another. 

Ex.   x  +  y<a''-^h\ 

It  is  to  be  remembered  that  any  positive  number  is  greater 
than  any  negative  number,  and  that  of  two  negative  numbers 
the  smaller  is  the  greater. 

Thus,  2  >  -  5 

-2>-3 

The  First  Member  of  an  inequality  is  the  expression  on 
the  left  of  the  sign  of  inequality ;  the  Second  Member  is  the 
expression  on  the  right  of  this  sign. 

170.  Two  inequalities  are  said  to  be  of  the  same  kind,  or 
to  subsist  in  the  same  sense,  when  the  greater  member  occu- 
pies the  same  relative  position  in  each  inequality ;  that  is,  is 
the  left-hand  member  in  each,  or  the  right-hand  member. 
Hence,  in  inequalities  of  the  same  kind  the  signs  of  inequality 
point  in  the  same  direction, 


Thus, 

xy2x- 
2x  +  \  ^  ^ 
3      >^'' 

-3 
-4 

are  of  the 

same 

kind; 

«j 

but 

2a>b- 

a 
2 

are  of  opposite  kinds. 

171.  Properties  of  Inequalities.    The  following  primary 
properties  of  inequalities  are  recognized  as  true : 

188 


INEQUALITIES.  189 

(1)  Adding  and  Subtracting  Quantities.  An  inequality 
jvill  be  unchanged  in  kind  if  the  same  quantity  be  added  to  or 
subtracted  from  each  member.     Hence, 

(2)  Terms  Transposed.  A  term  may  be  transposed  from  one 
member  of  an  inequality  to  the  other,  provided  its  sign  be  changed. 

(3)  Signs  Changed.  The  signs  of  all  the  terms  of  an  in- 
equality may  be  changed,  provided  the  sign  of  the  inequality  be 
reversed. 

(4)  Positive  Multiplier.  An  inequality  will  be  unchanged  in 
kind  if  all  its  terms  be  multiplied  or  divided  by  the  same  positive 
number. 

(5)  Raised  to  a  Po"wer.  An  inequality  will  be  unchanged  in 
kind  if  both  members  be  positive  and  both  be  raised  to  the  same  power. 

(6)  Inequalities  Combined.  If  the  corresponding  members 
of  two  inequalities  of  the  same  kind  be  added,  the  resulting  in- 
equality will  be  of  the  same  kind;  but  if  the  members  of  an 
inequality  be  siibtracted  from  the  corresponding  members  of 
another  inequality  of  the  same  kindj  the  resulting  inequality 
will  not  always  be  of  the  same  kind. 

172.  Application  of  Primary  Principles.  By  use  of  these 
primary  properties  complicated  relations  of  inequality  may 
be  reduced  to  simple  relations,  giving  more  or  less  definite 
results  of  value. 

Ex.  1.  Given  that  x  is  an  integer,  determine  its  value  from 
the  inequalities. 

r  4a;  -  7  <  2a;  +  3 
l3a;  +  l>13-a; 


f  2a;  <  10 
Transposmg  terms,         j 


4a;>12 
Dividing  by  coefficient  of  x  in  each  inequality, 

x<b 
a;>3 
• .  X  =  4,  ResuU. 


1 


190  ALGEBRA. 

Ex.  2.   Prove  that  the  sum  of  the  squares  of  any  two  un- 
equal quantities  is  greater  than  twice  their  product. 

Let  a  be  the  greater  of  the  two  quantities,  and  b  the  lessr 
Then, 

a-b>0 

r.(a-by>0 

,'.  a'-2ab  +  b'>0 

a'-i-b'>2ab. 
Ex.  3.   Prove  (a  +  6)  (6  +  c)  (a  +  c)  >Sabc. 
The  left-hand  member  when  expanded  becomes 

a{b^  +  c')  +  6(a2  +  c^)  +  cia"  +  b^)  +  2abc. 

But  from  Ex.  2,  a(62  +  d")  >  a{2bc) (1) 

bio"  +  c")  >  b{2ac) (2) 

c{a^  +  62)  >  c{2ab) (3) 

Also,  2a6c  =  2abc (4) 

Adding  (1),  (2),  (3),  (4), 

(a  +  6)  (6  +  c)  (a  +c)  >  Sabc. 

EXERCISE   64. 

Reduce — 

2.  (s-xy>(x-4y,  ^ 

3.  7ax  +  b>dax  +  5b.  6.  ^i:^>^Z:^- 

4a;  —  3       a;       3a;  +  8  ^  a -f  a;        64-a; 

3  2  21     '  '   a-x       2b-x' 

g   4(x  +  3)  ^8a;  +  37       7a;-29 
9       ""      18  5a;-12* 

Find  limits  of  x — 

9.  3a;+i>2a;  +  7.  10.  3(a;  -  4)  +  2  >  4(x  -  3). 

^-Kx  +  Q,  2(a;  +  l)<4(a;-l)-h3. 


INEQUALITIES.  191 

11.  What  number  is  that  whose  fifth  plus  its  sixth  is  greater 
than  6,  while  its  third  minus  its  eighth  is  less  than  4? 

12.  A  certain  integer  decreased  by  f  of  itself  is  greater  than 
J  of  the  number,  increased  by  5^ ;  but  if  ^  of  itself  be  added 
to  the  number,  the  sum  is  less  than  20.     Find  the  number. 

If  the  letters  employed  in  each  are  positive  and  unequal, 
prove : 

13.  3a''  +  6^^  >  2a(a  +  6).  ^^    ^^^-^2. 

14.  a'-h'>U'b-Zah\  ^^    ^  ,  T^  o  ^ 

17.  a  +  6>2]/a5. 

15.  a?-\-h^>a^b  +  ah\  is.  a^ ^h^  +  i^>ah^ ac^-hc. 

19.  6a6c  <  a(6'  -\-ah-\-  c^  +  cih"  +  6c  +  a'). 

20.  ah{a  +  6)  +  ac{a  +  c)  +  hc{h  +  c)< 2(a''  +  6'  +  c"). 

21.  a'  +  6'  +  c*  >  3a6c. 


CHAPTER    XV. 

INVOLUTION  AND   EVOLUTION. 

INVOLUTION. 

173.  Involution  is  the  operation  of  raising  an  expression 
to  any  required  power. 

Since  a  power  is  the  product  of  equal  factors,  involution  is 
a  species  of  multiplication.  In  this  multiplication  the  fact 
that  the  quantities  multiplied  are  equal  leads  to  important 
abbreviations  of  the  work. 

POWERS  OF  MONOMIALS. 

174.  Law  of  Exponents  or  Index  Law. 

Since  a^  =  aXaXaj 

(a'f  =  (a  X  a  X  a)  (a  X  a  X  a)  (a  X  a  X  a)  (aX  aX  a) 

In  general,  in  raising  a**  to  the  m'*  power,  we  have  the  factor 
a  taken  mXn  times,  or 

(a'*)«  =  a'"" L 

This  law  enables  us  to  abbreviate  the  process  of  finding  the 
power  of  a  factor  affected  by  an  exponent  into  a  mere  multi- 
plication of  exponents. 

Also,       (aby  =  ahXabXab to  n  factors 

=  (aXaXa to  n  factors)  (bXbXb to  n  factors) 

by  the  Commutative  Law  for  Multiplication. 

.  • .  (aby  =  arb'' 11. 

This  law  enables  us  to  reduce  the  process  of  finding  the 

192 


INVOLUTION.  193 

power  of  a  product  to  the  simpler  process  of  finding  the 
power  of  each  factor  of  the  product. 

175.  Law  of  Signs.     It  is  evident  from  the  law  of  signs 
in  multiplication  that — 

(1)  An  even  'power  of  a  quantity  (whether  plus  or  minus)  is 
always  positive. 

Exs.  (-3)''  =  9,        (-ahy  =  a*b\ 

(2)  An  odd  power  of  a  quantity  has  the  same  sign  as  the  orig- 
inal quantity. 

Exs.  (-ay  =  -a\        (+ay  =  a\ 

176.  Involution   of  Monomials  in  General.     Hence,  to 
raise  a  monomial  to  a  required  power, 

Raise  the  coefficient  to  the  required  power  ; 
Multiply  the  exponent  of  each  literal  factor  by  the  index  of  the 
required  power  ; 

Prefix  the  proper  sign  to  the  result. 

Ex.  1.   Find  the  cube  of  Zx^y. 

(Sx'yy  ^27xy. 

Ex.2.   (i-2aby=-B2a'b'\ 

177.  Powers  of  Fractions.     By  a  method  similar  to  that 
used  in  Art.  174,  it  can  be  shown  that 


\6'"/  ~  b^ 


Hence,  to  raise  a  fraction  to  a  required  power, 

liaise  both  numerator  and  denominator  to  the  required  power ^ 
and  prefix  the  proper  sign  to  the  resulting  fraction. 

16a"x* 


Ex./-^y 


6256y' 

18 


194 

ALGEBRA. 

' 

EXERCISE 

65 

, 

Write  the  square  of— 

1.  la'b. 

4.        =^^. 

5z' 

6.  -  13x»y. 

2.  -bxy\ 

3.  ixY. 

6a6 

'•    ir 

*      102 

8.  -42/«  +  ». 

9.         1. 

8cd 

Write  the  cube  of— 

10.  ^xy. 

11.  —2x\ 

12.  ^:^y\ 

13.  -SxY 

7x'" 

Write  the  value  of— 

16.  (7a6^c')l 

17.  (IWhJ. 

'•■(^J 

21.  (-2ar')'. 

22.  (-imO*. 

18.  (fx^2/)\ 

20.  (-JaV 

)*. 

23.  (3ia;'")^ 

24.*  (a' -25)1 

25.  (a  +  |)^ 

26.  (-x^-|2/)'. 

29. 
30. 
31. 
32. 

(a -26 +  3)'. 
(1-a  +  a^-ay 

(1  -  2x  +  la:^)'. 

28.  (|a:^  +  |2/z)^ 

33. 

(fx»-|x2/  +  i2/T. 

POWERS  OF  BINOMIALS. 

178.  General  Process.  In  obtaining  a  required  power  of 
a  binomial,  economies  are  possible  still  greater  than  those  used 
in  the  involution  of  a  monomial. 

It  is  sufficient  in  taking  up  the  subject  for  the  first  time  to 

*  For  this  and  the  succeeding  examples  in  this  exercise  see  Art.  181. 


INVOLUTION.  195 

obtain  several  powers  of  a  binomial  by  actual  multiplication, 
and,  by  comparing  them,  obtain  a  general  method  for  writing 
out  the  power  of  any  binomial.  A  formal  proof  of  the  method 
obtained  is  given  later. 

(a  +  hy  =  a'  +  Sd'b  +  Sab'  +  b\ 

(a  +  by  =  a*  +  4(f  6  -f  Qa'b'  +  4ab'  +  6*. 

(a  -f  bf  =  a'  +  ba'b  +  lOa'6'  +  lOa'6'  +  5a6*  +  b\ 

If  b  is  negative,  the  terms  containing  odd  powers  of  h  will 
be  negative ;  that  is,  the  second,  fourth,  sixth  and  all  even 
terms  will  be  negative. 

Comparing  the  results  obtained,  it  is  perceived  that 

I.  The  Number  of  Terms  equals  the  exponent  of  the  power 
of  the  binomial,  plus  one. 

II.  Exponents.  The  exponent  of  a  in  the  first  term  equals 
the  index  of  the  required  power,  and  diminishes  by  1  in  each 
succeeding  term.  The  exponent  of  b  in  the  second  term  is  1, 
and  increases  by  1  in  each  succeeding  term. 

III.  Coefficients.  The  coefficient  of  the  first  term  is  1 ; 
of  the  second  term  it  is  the  index  of  the  required  power. 

In  each  succeeding  term  the  coefficient  is  found  by  multi- 
plying  the  coefficient  of  the  preceding  term  by  the  exponent  of  a  in 
that  term,  and  dividing  by  the  exponent  of  h  increased  by  1. 

IV.  Signs  of  Terms.  If  the  binomial  is  a  difference,  the 
signs  of  the  even  terms  are  minus ;  otherwise  the  signs  of  all 
the  terms  are  plus. 

Ex.  (a  +  by  =  a'  +  7a'b  +  21a'b'  +  35a*6'  +  35a»6*  +  21a'6'^ 

-\-7ab'  +  b\ 
To  form  the  coefficient  of  the  third  term  we  have 


196  ALGEBRA. 

The  other  coefficients  are  determined  similarly.  It  is  to 
be  observed  that  the  coefficients  of  the  latter  half  of  the 
expansion  are  the  same  as  those  of  the  first  half  in  reverse 
order. 

179.  Binomials  with  Complex  Terms.  If  the  terms  of 
the  given  binomial  have  coefficients  or  exponents  other  than 
unity,  it  is  usually  best  to  separate  the  process  of  writing  out 
the  required  power  into  two  steps. 

Ex.  1.   Obtain  the  cube  of  2x  +  5y^. 
Since  (a  +  bf  =  a"  +  Sa'b  +  Sab'  +  b\ 
substituting  2x  for  a,  and  5y'  for  b, 

(2x  +  5^)^  =  (2x)'  +  3(2a;)^  (by')  +  Z(2x)  (SyJ  +  (ByJ 
=  8a;'  +  60a;y  +  ISOc?/*  +  125/. 

Ex.  2.   (2a;'  -  \yy  -  (2xy  -  4(2r')'  {\y')  +  ^(2xy  (\yy 

=  i6x^'^  -  8xy  +  f  xy  -  ixy  +  ,ji^2/'. 

180.  Application  to  Polynomials.  By  properly  grouping 
its  terms  a  polynomial  may  be  put  into  the  form  of  a  bino- 
mial, and  any  power  of  the  polynomial  obtained  by  use  of 
the  above  method  for  involution  of  a  binomial. 

Ex.  (a;  +  22/  +  Zzf  =  \(x  +  2y)  +  3z]' 

=  (a;  +  22/)'  +  3(a;  +  22/)'(3z)+3(a;  +  22/)(32)= 

+  (3z)' 
=  «» -f  Gx'^y  +  12x2/' +  82/' +  9x'z  H- 36a;2/z 
+362/'z  +  27a;z'  +  bAyz'  +  27z'. 

181.  Cases   of  Involution  Previously  Considered.     In 

Arts.  85,  86,  89  important  special  cases  of  involution  have, 
been  considered,  and  should  be  here  recalled: 

1.  The  square  of  the  sum  or  difference  of  two  quantities; 

2.  The  square  of  any  polynomial. 


J<!  VOLUTION. 

rrt^rtfl" 

EXERCISE  66. 

iLLfcilUJ. 

1.  ia-b)\ 

2.  (x  +  1)». 

3.  (1-^)*. 

4.  (a -2/. 

5.  (2  +  x')*. 

6.  (a -26/. 

14.  (3-iO^ 

7.  (^  +  3/. 

17.  (ar'  +  x-l)*. 

8.  (a' -26)'. 

18.  (x'-Sx-iy. 

9.  (_2c-dy. 

19.  (a'  +  ac  +  O*. 

10.  (a -36')* 

20.  (a:-2/  +  2)'. 

11.  (7  -  3x').' 

21.  (2x'-a:  +  3)». 

12.  (x2/'  +  2)*. 

22.  (l+x-x^y. 

EVOLUTION. 

197 


182.  The  Root  of  a  quantity  is  that  quantity  which  taken 
as  a  factor  a  given  number  of  times  will  produce  the  given 
quantity. 

183.  Evolution  is  the  process  of  finding  a  required  root  of 
a  quantity. 

Hence  evolution  is  the  inverse  of  involution. 

184.  The  Radical,  or  Root  Sign,  it  will  be  recalled,  is  |/. 
The  number  indicating  the  required  root  is  written  as  a  small 
figure  over  the  radical  sign  and  is  called  the  index  of  the 
root.     For  the  square  root  the  index  number  is  omitted. 

Thus,  Va,  Va,  {^'a,  indicate  the  square  root,  the  cube  root, 
and  the  n'*  root  of  a  respectively. 


198  ALGEBRA. 

EVOLUTION  OP  MONOMIALS. 

185.  Index  Law.     Since  (cry^aT''  (Art.  174)  it  follows 
that  _  • 

^~d^=ar I. 

where  m  and  n  are  positive  integers. 

This  reduces  the  process  of  finding  the  root  of  a  quantity 
affected  by  an  exponent  to  a  division  of  exponents. 

Also,  \/ah  ~  ■\ya{/l> II. 

For  let    v^'a  =  Xy        yl)  —  y ; 

.•.a:"  =  a,...(l)        y-  =  h...(2) 
But  xV  =  (P^yT  (by  Art.  174) 

Substitute  for  a;«  and  2/"  fro-  (1)  and  (2), 

«^  =  (v^ai/F)« (3) 

Extract  the  n'*  root  of  each  member  of  (3), 

This  reduces  the  process  of  finding  the  n'*  root  of  a  product 
to  the  simpler  process  of  finding  the  root  of  each  factor. 

186.  Law  of  Signs.     From  the  law  of  signs  for  multipli- 
cation it  follows  that — 

(1)  Any  even  root  of  a  positive  quantity  may  be  either  positive 
or  negative. 

Ex.  1/9  =  +  3,  or  -  3. 

It  is  convenient  for  the  present  to  consider  only  positive 
roots  of  even  powers. 

(2)  No  negative  quantity  can  have  an  even  root. 

Ex.  The  square  root  of  —  4  is  neither  +  2  nor  —  2^  since 
neither  of  these  multiplied  by  itself  will  give  —  4, 

(3)  The  odd  root  of  a  quantity  has  the  same  sign  as  the  quantity 
itself. 

Ex.      1^=27^=- 3. 


SQUARE  ROOT.  199 

187.  Entire  Process.    Hence,  to  extract  a  required  root  of 
any  monomial, 

Extract  the  required  root  of  the  coefficient ; 
Divide  the  exponent  of  each  letter  by  the  index  of  the  required 
root ; 

Prefix  the  proper  sign  to  the  result. 


EXERCISE  67. 

Write  the  square  root  of — 

1.  9xy.  4.  16a;y.  6.  |a;Y'. 

2.  25a*.  36a^^  121aV" 

3.  144/".  *  49a;'***  *   812/'"  +  ^ 

Write  the  cube  root  of— 


27a"a;'  iQ    8a;V 


n  +  8 


12.  --— —  •  14. 


1000 


10.  -ia'6^  •        3432^9  *       a;**"* 

Write  the  value  of — 

15.  \'-bl2x\  18.  VM^^'^  21     *I^25P 

16.  1^16^^.  19.  V7=^.  •   \2/*"  +  «' 

17.  l^^^^^y:  20.  V^^P-  22.    l^-Aa^2/*- 

23.  l/25ar'  +  20a;  +  4.  25.   l/a^6^  -  f6a6c  +  64c'. 

24.  l/9a;*-42a;-^  +  49.  26.   Vl  +  18xy  +  Sl^y. 

SQUARE  ROOT. 

188.  Square  Root  of  Polynomials.  Our  object  is  to  dis- 
cover such  a  relation  between  the  terms  of  a  binomial  (or  in 
general  of  a  polynomial)  and  the  terms  of  its  square  (as,  for 
instance,  between  a  -\-h  and  its  square,  a^  +  2ah  +  6^),  that  we 


200  ALGEBRA. 

can  state  this  relation  in  the  inverse  form  as  a  general  method 
for  readily  determining  the  square  root  of  any  polynomial 
which  is  a  square. 


a"  +  lab  +  h'' 


The  first  term  of  the  root,  a,  is  the  square 
root  of  the  first  term,  d^^  of  the  square  ex- 


2a  +  6    2a6  +  IP"  pression.    The  second  term  of  the  root,  6, 

2a6  +  IP'  occurs  in  the  second  term,  2a6,  of  the  square 

^  expression,  and  may  be  obtained  frcftn  it  by 

dividing  by  twice  the  first  term,  or  la  (called  the  trial  divisor).  If  we  take 
la  and  add  h  to  it  (giving  2a  +  6,  called  the  complete  divisor),  and  multi- 
ply the  sum  by  6,  we  get  2a6  +  6^,  which  is  the  rest  of  the  square  expres- 
sion after  a^  has  been  subtracted.  This  last  step,  therefore,  furnishes  a  test 
of  the  accuracy  of  the  work. 

Ex.  Extract  the  square  root  of  16x^  —  2Axy  +  92/'* 


16x2 


^x-Zy 


-  24xy  +  9y^ 

-  lAxy  +  9.^2 


Taking  the  square  root  of  the  first  term,  \Qx^,  we  obtain  Ax,  which  is 
placed  to  the  right  of  the  given  expression  as  the  first  term  of  the  root. 
Subtract  the  square  of  4x  from  the  given  polynomial. 

Taking  twice  the  first  term  of  the  root,  8x,  as  a  trial  divisor,  and  dividing 
it  into  the  first  term  of  the  remainder,  we  obtain  the  second  term  of  the 
root,  —  Zy.  This  is  annexed  to  the  first  term  of  the  root  and  also  to  the 
trial  divisor  to  make  the  complete  divisor,  8x  —  Zy. 

189.  Square  Root  to  Three  or  More  Terms.  In  squaring 
a  trinomial,  a  +  6  +  c,  we  may  regard  a-\-h  as  a  single  quan- 
tity, and  denote  it  by  a  symbol,  as  ^,  and  obtain  the  square 
in  the  form  p^  +  2pc  +  cl 

Evidently  we  may  reverse  this  process,  and  extract  a  square 
root  to  three  terms,  by  regarding  two  terms  of  the  root  when 
found  as  a  single  quantity.  So  a  fourth  term  of  a  root,  or 
any  number  of  terms,  may  be  found  by  regarding  in  each 
case  the  root  already  found  as  a  single  quantity. 


SQUARE  ROOT.  ^  201 

Ex.  Extract  the  square  root  of  x*  —  6af'  +  19x'  -  30x  +  25. 

^  -  6^3  +  19a;2  -  30a;  +  25  |  a;'  -  3a;  +  5 
a;* 


2a;2  -  3a;  I  -  6a;^  +  19a;2 
-  fia;^  +    9a;2 


2a;2  -  6a;  +  5 


+  lOa;-''  -  30a;  +  25 
+  ]0a;2  -  30a;  +  25 


The  first  two  terms  of  the  root,  x^  -  3a;,  are  found  as  in  the  example  in 
Art.  188. 

To  continue  the  process,  we  consider  the  root  already  found,  x^  —  3a;,  as 
a  single  quantity,  and  multiply  it  by  2  to  make  it  a  trial  divisor. 

Dividing  the  first  term  of  the  remainder,  lOa;^,  by  the  first  term  of  the 
trial  divisor,  +  2a;'^,  we  obtain  the  next  term  of  the  root,  +  5. 

The  process  is  then  continued  as  before. 

Hence,  in  general,  to  extract  the  square  root  of  a  poly- 
nomial, 

Arrange  the  terms  according  to  the  -powers  of  som»  letter  ; 

Extract  the  square  root  of  the  first  term,  set  down  the  result  as 
the  first  term  of  the  root,  and  subtract  its  square  from  the  given 
polynomial ; 

Take  twice  the  root  already  found  as  a  trial  divisor,  and  divide 
it  into  the  first  term  of  the  remainder  ; 

Set  down  the  quotient  as  the  next  term  of  the  root,  and  also  annex 
it  to  the  trial  divisor  to  form  a  complete  divisor  ; 

Multiply  the  complete  divisor  by  the  last  term  of  the  root,  and 
subtract  the  product  from  the  first  remainder  ; 

Continue  the  process  till  all  terms  of  the  root  are  found. 

EXERCISE   68. 

Find  the  square  root  of— 

1.  x'  -  4a;'  +  6a;''  -  4a;  +  1. 

2.  l-2a-a'  +  2a'  +  a\ 

3.  9a;*  -  12a:'  +  lOx'  -  4a;  +  1. 

4.  25  +  30a;4-19a:'  +  6a;'  +  a;*.      • 


202  •  ALGEBRA, 

6.  n'  —  4n^  +  4ri*  +  Qtv'  -  12n'  +  9. 

6.  ^x^  +  12x'  +  a:*  -  24a:='  -  14x'  +  12a;  +  9. 

7.  1  +  IGm"  -  407?i*  +  10m  -  Sm'^  +  25m^ 

8.  46?i'  +  25n^  +  4n'  +  25  -  44n'  -  40n  -  12n^ 

9.  9a;'  +  92/'  +  24x^2/  +  24:X^  -  S^y  -  8xy  -  bO^^y", 

10.  m'  +  9  +  x'  +  6m  +  6a;  +  2ma;. 

11.  1  +  5x^  +  2a;*  +  a;'  -  4a;'  +  2a;-''  +  2x. 

12.  28a;'  -  47a;*  +  49x'  -  42a;'  -  4a;'  +  16a;  +  4. 

13.  \x''  -  bx  +  25.  16.  x*  +  2a;'  -  a;  +  \. 

14.  |x'  -  5a;2/  +  ^y\  17.  >*  -  Ja'  +  -^a'  -  4a  +  36. 

42/''        2/  a;''        a;  a         a 

19.  ^x'  -  la;'  +  W^^'  -  3a;  +  ^. 

20.  ^x*-|-ar'  +  |K--|a;  +  f|. 

21.  l  +  a-Aa'-K-|a*  +  t«'  +  «*'- 
r*  r^  1  ^'^        1 

^^•T  +  -  +  7  +  i  +  &  +  7 

23.  — -aa;  +  — -2  +  —  +—. 
x^  a'  a;         4 

Find  to  three  terms  the  square  root  of — 

24.  1  +  4a;.  27.  a^  -\*4h.  30.  a;"  —  1. 

25.  1  — 2a.  28.  9a'  — 4a;.  31.  a;' +  3. 

26.  a;'  -  6.  29.  4a'  -  6a6.  32.  a'  +  3a6  -  2h\ 

190.  Square  Root  of  Arithmetical  Numbers.  The  same 
general  method  as  that  used  in  Art.  188  can  be  used  to  extract 
the  square  root  of  arithmetical  numbers.  The  details  of  the 
process,  however,  are  somewhat  different,  owing  to  the  fact 
that  all  the  numbers  which  compose  a  given  square  number 
are  given  united  as  a  single  number. 


SQUARE  ROOT.  203 

Thus,  (43)'  =  (40  +  Zy  =  1600  +  240  +  9  -  1849. 

Hence,  given  1 849  to  extract  its  square  root,  the  square  of  the  first  num- 
ber, 1600,  is  not  presented  explicitly  as  it  would  be  in  an  algebraic  expres- 
sion, but  must  be  determined  indirectly. 

The  first  step  is  to  mark  off"  the  figures  of  the  given  number  whose  root  is 
to  be  extracted  into  periods  of  two  figures  each,  beginning  at  the  decimal 
point,  and  then  to  determine  the  largest  square  number  represented  in  the 
first  period  of  figures  at  the  left  as  a  trial  number.  If  the  first  figure  of  the 
root  be  in  the  tens'  place,  and  therefore  followed  by  one  zero  (as  4  in  40 
above),  its  square  will  be  followed  by  two  zeros,  as  in  1600.  If  the  first 
figure  of  the  root  had  been  in  the  hundreds'  place,  and  therefore  followed 
by  two  zero^,  its  square  would  have  been  followed  by  four  zeros ;  that  is, 
there  are  two  additional  zeros  in  the  square  for  each  additional  zero  follow- 
ing the  first  figure  of  the  root.  Hence  comes  the  significance  of  separating 
the  given  square  number  into  periods  of  two  figures  each,  and  extracting 
the  approximate  square  root  of  the  left-hand  period  of  figures. 

We  will  illustrate  by  an  example,  using  the  algebraic  formula  (a  +  by 
=  a^  +  2ab  +  b"^,  to  show  the  essential  identity  of  the  arithmetical  and 
jilgebraic  processes. 


Ex.  Extract  the  square  root  of  1849. 


Trial  divisor,  2a  =  80 
b=    3 


1849. 1  40 +  3 
1600 

249 

249 


Complete  divisor,  2a  +  6  =  83 
This  work  may  be  put  in  the  following  abbreviated  form : 

1849. 1  43 
16 

83  I  249 

I  249 

191.  Square  Root  of  Decimal  Numbers.  If  it  be  required 
to  extract  the  square  root  of  a  decimal  number,  as  28.09.  we 
may  proceed  thus : 

\100        1/100       10 
It  is  better,  however,  to  put  this  work  into  a  different  form 


204  ALGEBRA. 

by  marking  off  the  given  number  into  periods  of  two  figures 
each,  beginning  at  the  decimal  point  and  marking  both  to  the 
right  and  left.  If  necessary  annex  a  zero  to  complete  the  last 
period  of  figures  to  the  right ;  in  such  cases,  however,  the  root 
cannot  be  exactly  extracted. 

Ex.  Extract  the  square  root  of  18.550249. 

18.550249  I  4.307,  Boot. 

16 

83 


255 
249 


8607 


60249 
60249 


192.  Square  Root  of  Comraon  Fractions.  If  the  de- 
nominator of  the  fraction  whose  square  root  is  to  be  extracted 
is  a  perfect  square,  extract  the  root  of  the  numerator  and  de- 
nominator separately  and  divide  the  one  result  by  the  other. 

^         1289      V2m      17 

iliX.    \  —  — =  —  * 

^324      1/324      18 

If  the  denominator  is  not  a  perfect  square,  reduce  the  frac- 
tion to  a  decimal  and  extract  the  root  of  the  decimal. 

Ex.    Vi  -  T/0.66666666  + 

0.66666666+  |  0.8164+^ 
64 


161 

266 
161 

1626 

10566 
9756 

1632 

4 

81066 
65296 

Hence,  in  general,  to  extract  the  square  root  of  an  arith- 
metical number, 


SQUARE  ROOT.  205 

Separate  the  number  into  periods  cf  two  figures  each,  beginning 
at  the  decimal  point ; 

Find  the  greatest  square  in  the  left-hand  period,  and  set  down  its 
root  as  the  first  figure  of  the  required  root ; 

Square  this  figure,  subtract  the  result  from  the  left-hand  period, 
and  to  the  remainder  bring  down  the  next  period  ; 

Double  the  root  already  found  for  a  trial  divisor,  divide  it  into 
the  remainder  {omitting  last  figure  of  the  remainder),  and  annex 
the  quotient  obtained  to  the  root  and  also  to  the  trial  divisor. 

Multiply  the  complete  divisor  by  the  figure  of  the  root  last  found, 
and  subtract  the  result  from  the  remainder  ; 

Proceed  in  like  manner  till  all  the  periods  of  figures  have  been 
used. 

EXERCISE   69. 

Find  the  square  root  of — 

1.  7225.  6.  337561.  11.  199.204996. 

2.  2601.  7.  567009.  12.  10.30731025. 

3.  8464.  8.  11573604.  18.  254046.2409. 

4.  105625.  9.  36144144.  14.  .0291419041. 

5.  182329.  10.  8114.4064.  15.  1513689.763041 

Find  to  four  decimal  places  the  square  root  of — 

16.  7.       19.  ^.  '22.,  ^.  25.  .049. 

17.  11.      20.  2^.      23.  1|.      26.  1.0064. 

18.  12.5.     21.  0.9.      24.  ^.      27.  36^. 

Compute  to  three  decimal  places  the  value  of — 


jT 


28.  \/2+VE.  32.  l/2V7-f-3l/2.  l5(V^-T/^_ 

29.  l/ 1/5-1.  33.  1/31/6-21/7.  *                 2 

80.  \/VW-V%.  34.  JV5-1/2  36.  j7,V^4-2l/5^ 

81.1/31/3+1/5.                      4  ^ 


a' 

3a' 

+  Zah  +  6' 

+  Sa'^ft  +  8a62  +  6' 

Za?  +  3a6  +  b" 

+  Za'b  +  3a62  +  6' 

206  ALGEBRA. 

CUBE  ROOT. 

193.  Cube  Root  of  Polynomials.  Our  object  is  to  deter- 
mine such  a  relation  between  the  terms  of  a  binomial,  or,  in 
general,  of  a  polynomial,  and  the  terms  of  its  cube  (as  between 
a  +  ft,  and  its  cube,  o?  +  3a^6  +  3aft^  +  6^),  that  we  may  be  able 
to  state  this  relation  in  the  inverse  form  as  a  general  method 
for  determining  the  cube  root  of  any  polynomial  which  is  a 
perfect  cube. 

a'  +  Zo}b  +  3a62  +  53  [0^  +  5  The  first  term  of  the 

root,  a,  is  the  cube  root 
of  the  first  terra,  a', 
of  the  cube  expression. 
The  second  term  of  the 
root,  6,  occurs  in  the 
second  term  of  the  cube  expression,  3a^6,  and  may  be  obtained  from  it  by 
dividing  it  by  Za^ ;  that  is,  by  three  times  the  square  of  the  first  term  of 
the  root  (called  the  trial  divisor).  If  we  take  the  trial  divisor,  and  add  to 
it  three  times  the  product  of  the  first  term  of  the  root  by  the  second  term, 
3a6,  and  also  the  square  of  the  second  term  of  the  root,  h"^,  we  get  3a' 
+  3a6  4-  6^  (called  the  complete  divisor)  ;  this  multiplied  by  the  second 
term  of  the  root  gives  Za^b  +  Zab^  +  6^,  the  rest  of  the  cube  expression 
after  a'  has  been  subtracted. 
This  last  step,  therefore,  furnishes  a  test  of  the  accuracy  of  the  work. 

194.  Three  or  More  Terms  in  the  Root.  In  cubing  a 
trinomial,  a  +  6  +  c,  we  may  regard  a  +  b  as  a  single  quan- 
tity, and  denote  it  by  p,  and  obtain  the  cube  in  the  form 
p^  +  3p'c  4-  3pc^  +  (f.  Evidently  we  may  reverse  this  process, 
and  extract  a  cube  root  to  three  terms,  by  regarding  two  terms 
of  the  root  when  found  as  a  single  quantity.  So  a  fourth 
term  or  any  number  of  terms  of  a  root  may  be  found  by 
regarding,  in  each  case,  the  root  already  found  as  a  single 
quantity. 

We  will  now  extract  the  cube  root  of  a  polynomial  expres- 
sion indicating  at  each  step  the  trial  divisor  and  complete 
divisor. 


CUBE  ROOT. 


207- 


g 

fL 

o 

^' 

;2. 

OS  g 

1^ 

g 

+  iS 

^* 

I«-f 

5 
-< 

^  +  ►^ 

B 

1       00    ^ 

za  >^ 

C/,  ^   S" 

fs    II 

V  .,"  ^ 

^.   CO 

"  is: 
5  i* 

I 

f5 

tr 

a> 

o 

S 

(T 

o 

i_^ 

Q 

Q 

<rt- 

O 

•->» 

00 

«• 

+ 

1? 

CO 

+ 

%. 

CO 

1 

Oi 

I 

^ 

f 

1 

1 

1 

OS 

1 

^ 

I— »• 

at 

1 

CO 

1 

n. 

CO 

+ 

^ 

I—'' 

+ 
t— > 

K 

? 

+ 

+ 
fcO 

1 

^i 

i 

1 

t— I 

to 

to 

9^ 

Ci 

€ 

+ 

1 

1 

208  ALGEBRA, 

EXERCISE  70. 

Find  the  cube  root  of — 

1.  a' +  Qa'x  +  12ax' +  Sx\ 

2.  27-27a  +  dd'-a\ 

3.  l-12a:  +  48x^-64rl 

4.  a^  -  Sa'  -  3a*  +  Ha'  +  6a'  -  12a  -  8. 
6.  a:«  -  3x^  +  6a:*  -  7a:'  +  6a:'  -  3a:  +  1. 

6.  1  -  9a:  +  21a:'  +  9a:'  -  42a:*  -  36x^  -  8a:«. 

7..  12a;*  -  36a:  +  64a:'  -  6a:'  -  8  +  1 17a:'  -  144a:l 

8.  95a'  +  72a*  -  72a'  +  15a^  +  15a  +  a'  -  1. 

9.  114a:*  -  171a:'  -  27  -  135a:  +  8a;«  -  60a:^  +  55ar». 
10.  8  +  27n'  -  36?i  -  81n^  +  907i'  -  135n'  +  13571*. 


11. 


a:' 
8  ' 

x' 
4 

-i 

1 

27* 

-i- 

2y 

22/ 
3 

18. 

0?- 

•  3a;'  +  6a:  -  7  + 

6_ 

X 

hi- 

14. 

1  + 

3_ 
a 

a'       a' 

18      27      27 
a*       a'       a' 

■ 

15. 

x'-V 

y 

15a:* 

2f 

45a:' 

,  27_^_ 

27 

10i» 

42/* 

+  2y 

¥" 

2/" 

27ar^ 


195.  Cube  Root  of  Arithmetical  Numbers.  The  same 
general  method  as  that  used  in  Art.  194  can  be  used  to 
extract  the  cube  root  of  arithmetical  numbers.  As  in  square 
root,  the  process  is  slightly  different  from  the  algebraic  one, 
owing  to  the  fact  that  all  the  numbers  which  compose  a  given 
cube  are  given  united  or  fused  into  a  single  number. 

-  Thus,    (42)=*  =  (40  +  2)='  =  4a''  +  3  X  402  X  2  +  3  X  40  X  22  +  2^ 
=  64000  +  9600  +  480  +  8 
=  74088. 
Hence,  given  74088,  to  extract  its  cube  root,  the  cube  of  the  first  number, 


CUBE  ROOT.  209 

or  64000,  is  not  given  explicitly,  as  it  would  be  in  an  algebraic  expression, 
but  must  be  determined  indirectly.  This  is  done  by  marking  off  the  given 
cumber  into  periods  or  groups  of  three  figures  each,  beginning  at  the  deci- 
mal point,  and  then  determining  the  largest  cube  represented  in  the  first 
period  of  figures,  and  taking  its  cube  root  as  a  trial  number  for  the  first 
figure  of  the  root.  The  reason  for  marking  off"  the  given  number  into 
periods  of  three  figures  each  may  be  briefly  stated  thus :  If  the  first  figure 
of  the  root  be  in  the  tens'  place  and  therefore  followed  by  one  zero  (as  40 
above),  its  cube  will  be  followed  by  three  zeros  (as  64000).  If  the  first 
figure  of  the  cube  root  be  in  the  hundreds'  place,  and  therefore  followed  by 
two  zeros,  its  cube  would  be  followed  by  six  zeros.  For  every  additional 
zero  in  the  root  there  are  three  additional  zeros  in  the  cube.  Hence  arises 
the  significance  of  separating  the  given  number  into  periods  of  three  figures 
each,  and  extracting  the  approximate  cube  root  of  the  left-hand  period. 

We  will  now  illustrate  the  general  process  of  extracting  an  arithmetical 
cube  root,  using  the  algebraic  formula  (a  +  6)'  =  a?  -\-  Za^b  +  ^ab"^  +  6', 
to  show  the  essential  identity  of  the  arithmetical  and  algebraic  processes. 

Ex.   Extract  cube  root  of  74088. 


Trial  Divisor, 
Complete  Divisor, 


74088  I  40  +  2 

a3  =  403  =  64000 

Za^=       3  X  40^  ■■=  4800 

"TU08B 

Zah  =  3  X  40  X  2  =  240 

62  =       22  =   4 
3a2  +  Zab  +  6^  =  5044 

10088 

m  of  work— 

74088  [42 
64 

3  X  402  =  4800 

10088 

3  X  40  X  2  =  240 

22=   4 

5044 

10088. 

196.  Cube  Root  of  Decimal  Numbers  and  Fractions. 

For  a  reason  similar  to  that  given  in  Art.  191  for  square  root 
of  decimal  numbers,  in  extracting  the  cube  root  of  decimal 
numbers  we  mark  off  the  decimal  numbers  into  periods  of 
three  figures  each,  beginning  at  the  decimal  point,  and  sup^ 

14 


210  ALGEBRA. 

plying   a  sufficient  number  of  zeros   when  the  right-hand 
period  is  incomplete. 

Ex.  1.   Extract  the  cube  root  of  130.323843. 

130.323843  |  5.07 

125 

Trial  Divisor  =  3  x  (500)^  =  750000    5323843 
3  X  500  X  7  =    10500 


49 


Complete  Divisor  =  760549 


5323843 


Ex.  2.   Extract  the  cube  root  of  3%  to  4  decimal  places. 


^-0.416666666666+. 


0.416666666  +  |  0.7469+ 
343 

3  X  (70)2  _  14700 


3  X  (70  X  4)  =   840 
42=   16 


15556 

3  X  (740)2  _  1642800 

3  X  (740  X  6)  =   13320 

62  =     36 


73666 


62224 


11442666 


9936936 


1656156 

3  X  C7460)2  =  166954800  I  1505730666 

I  1502593200 

3137466 

The  first  three  figures  of  the  root  are  found  directly.  The  last  figure  is 
then  found  by  division  of  the  remainder,  using  three  times  the  square  of 
the  root  already  found  as  a  divisor.  The  number  of  figures  of  the  root  that 
may  thus  be  found  by  division  is  two  less  than  the  number  of  figures 
already  found. 

Hence,  in  general,  to  extract  the  cube  root  of  an  arithmetical 
number, 

Separate  the  number  into  periods  of  three  figures  each,  beginning 
at  the  decimal  point ; 

Find  the  greatest  cube  in  the  left-hand  period,  and  set  down  its 
cube  root  as  the  first  figure  of  the  required  root; 


CUBE  BOOT.  211 

Cube  this  figure,  and  subtract  the  result  from  the  left-hand 
period,  and  annex  the  next  period  of  figures  to  the  remainder; 

Take  three  times  the  square  of  the  root  already  found  with 
zero  annexed,  as  a  trial  divisor;  divide  the  remaind^'r  by  it^ 
and  set  doivn  the  quotient  as  the  next  figure  of  the  root; 

Complete  the  trial  divisor  by  adding  to  it  three  times  the  product 
of  the  first  figure  of  the  root  with  zero  annexed,  midtiplied  by  the 
last  figure,  and  the  square  of  the  last  figure ; 

Multiply  this  complete  divisor  by  the  figure  of  the  root  last  foundj 
and  subtract  the  result  from  the  remainder  ; 

Proceed  in  like  manner  till  all  the  periods  have  been  used. 

EXERCISE   71. 

Find  the  cube  root  of— 

1.  3375.  4.  43614208.  7.  344324.701729. 

2.  753571.  6.  32891033664.         8.  .000127263527. 

3.  1906624.  6.  520688691.125.      9.  0.991026973. 

Find  to  three  decimal  places  the  cube  root  of — 

10.  75.  12.  5.6.  14.  7^^.  16.  ^.  18.  1^. 

11.  6.  13.  3f.  15.  19f         17.  y^.        19.  8^. 

Compute  the  value  of — 

20.  Vb  +  2V%.  21.  I>'3l/10-2vl8. 


22.  v/3l70:8-2l/1.93'5. 

197.  Higher  Roots  Obtained  by  Successive  Extractions. 

By  the  law  of  exponents  the  square  of  the  square  of  any 
quantity  gives  the  fourth  power  of  the  quantity.  Hence,  re- 
versing the  process,  the  fourth  root  of  a  quantity  is  the  square 
root  of  the  square  root  of  the  quantity.  Similarly,  the  sixth 
root  of  a  quantity  is  the   square  root  of  the  cube  root  of  the 

quantity.     The  eighth,  ninth,  tenth roots  of  a  quantity 

may  be  found  by  similar  methods. 


212  ALGEBRA. 

Ex.   Extract  the  fourth  root  of 

81a*  +  lOSa"  +  54a'  +  12a  +  1. 

Obtain  first  the  square  root  of  the  given  expression,  which 
is  9a'  +  6a  +  1.  Extracting  the  square  root  of  this,  we  obtain. 
3a  +  1,  the  fourth  root  of  the  original  expression.  -' 

EXERCISE   72. 

Find  the  fourth  root  of — 

1.  130321.  2.  3418801.  3.  90.  4.  0.8. 

6.  1  -  12ah  +  54a'6'  -  108a'6'  +  Sla'b\ 
6.  a:*  -  2x'  +  ^x"  -ix  +  ^, 

*     2/*  y  ^'        16a;* 

8.  64x'  -  56a;*  +  16a;'  +  a;«  +  16  -  323;^  +  IQx'  -  Sx'  +  64.'5. 

Find  the  sixth  root  of — 

9.  7529536.  10.  1544804416.  11.  15. 

12.  x'  4-  1215a;'  +  729  -  1458a;  +  135a;*  -  540a;'  -  183;^. 

13.  64a«  -  192a*  +  ^-^  +  -^-160  +  240a'. 

a'       a*       a* 

14.  4096x^»  -  3072a;^°  +  960a;«  -  160x«  +  15a;*  -  fa;'  +  j^. 


CHAPTER    XVI. 
EXPONENTS. 

198.  Positive  Integral  Exponents.  Using  a'  as  a  brief 
symbol  for  aXaXa,  and  a"*  as  a  brief  symbol  for  aXa 

XaX  a torn  factors,  we  have  already  found  the 

following  laws  to  govern  the  use  of  positive  integral  expo- 
nents : 

I.  a'"Xa"=:a"*  +  ». 

11.  —  =  ar-''A£  m>n. 

III.  (a"*)"  =  a*"". 

IV.  ^y^=fjm^ 

V.  (ahy  =  orh\ 

199.  Fractional  and  Negative  Exponents.  Just  as  by 
using  fractions  as  well  as  integers,  and  negative  as  well  as 
positive  quantity,  the  field  of  quantity  and  operation  in  al- 
gebra is  greatly  extended,  some  processes  made  simpler,  and 
others  more  powerful,  so  by  introducing  fractional  and  neg- 
ative exponents  we  get  like  results. 

As  fractional  and  negative  exponents  have  no  meaning 
belonging  to  them  at  the  outset,  it  will  be  most  advanta- 
geous to  sui)pose  that  the  first  and  fundamental  Index  Law, 
a™  X  a"  =  a*"  ^  '*,  holds  for  fractional  and  negative  exponents, 
and  then  inquire  what  meaning  must  be  assigned  to  these 
exponents.  We  limit  the  fractional  and  negative  exponents 
here  treated  to  those  whose  terms  are  either  positive  or  neg- 
ative integers,  and  commensurable;  that  is,  expressible  in 
terms  of  the  unit  of  quantity  used  in  the  given  problem. 

213 


214  ALGEBRA. 

Thus,  exponents  like  1/2,  as  in  a^^  are  not  included  in  the 
iiscussion,  though  the  student  will  find  later  that  the  same 
laws  hold  for  these  exponents. 

200.     I.  Meaning  of  a  Fractional  Exponent. 
Since  by  Index  Law  I., 

it  follows  that  a^  is  one  of  the  three  equal  factors  which  may 

2 

be  considered  as  composing  a^ ;  that  is,  a^  is  the  cube  root  of  al 

2. 

Hence,  in  the  exponent  of  a^,  the  numerator,  2,  denotes 
the  power  of  a  to  be  taken,  and  the  denominator,  3,  denotes 
the  root  of  this  power  to  be  extracted. 

So,  in  general, 

p       p        p       p 
a^  X  a'^  X  a"  X  a'^ to  q  factors 

?.  +  ?.+?.+ to  3  terms. 

=  a«      =--oF. 

Hence,  in  general,  in  a  fractional  exponent  the  numerator 
denotes  the  power  of  the  base  that  is  to  be  taken,  and  the  denomina- 
tor denotes  the  root  that  is  to  be  extracted. 


Ex.  1. 

83  =  1^8^=1^63=4. 

Ex.  2. 

aixaixai  =  ai'i'i 

=  aT2"  Product 

Ex.  3.  x^^'x^  =  x^Xx^. 

=  a;^  Product, 
Ex.4.  32*=  1^^  =  2^  =  64. 


EXPONENTS.  215 


EXERCISE  73. 

Express  with  radical  signs — 


1.  ai 

4.  2aK 

7.  a^m*. 

10. 

s 

2.  x^. 

5.  ax^. 

8.  5x^2/^- 

11. 

by\ 

3.  oi 

6.  2a^6^. 

9.  2c*ci* 

12. 

n     m 
^my2n^ 

Express  with  fractional  exponents — 

13.  V^. 

14.  1/?. 

16.  aVx. 

17.  hv'y. 

19.  1/5  iV. 

20.  21^x^1/^. 

22. 
23. 

4l^al/?'' 
2i^STv'3' 

15.  2l^P. 

18.  2x1/^. 

21.  1^5  V>^. 

SVTVW 

Find  the  value  of— 

24.  27^. 

27.  v' W"       ' 

■      30.  (-27)* 

33. 

(-243)* 

25.  25* 

28.  ^  64\ 

31.  (-32)* 

34. 

(^)*. 

26.  16^. 

29.  (-8)*. 

32.  (-216)^ 

36. 

(!i)* 

Simplify  the  following  by  performing  the  indicated  opera- 
tions : 

36.  a^Xa^.  40.  2*a;*  X  2^.  44.  Vcc^^a?. 

37.  2a  X  a*.  41.  J  Va\  45.  ^'2VT. 

38.  a'x2Xa%^.        42.  7l/av^a^.  46.  a^y'^-x^Va. 

39.  3x^  X  a%3.         43.  2x'|>^^.  47.  a;*|XF^  •  2x^. 

i^y-^      2xiv^  V2^'l  2*7* 


a^  »/  a;-      zrc*  y  a'  V2-^'l 

48.    -—-  X  ——-'  49.  -— V^  X  i: 


^^  iTc  *  i/3|>-5      1^3V5« 


216  ALGEBRA. 

201.  n.  Meaning-  of  the  Exponent  Zero,  or  of  a\ 

By  the  Index  Law  I., 

a'  X  a""  =  a'  +  "^  =  or  ==1  X  or 
.  • .  a"  =  1. 

Hence,  aP  is  another  symbol  for  unity. 
The  student  will  realize  the  meaning  of  aP  more  readily 
thus, 

a™ 
By  direct  division,  —  =  1 

^  or 

By  subtraction  of  exponents,  —  =  o^ 
.-.byAx.  1,  a'  =  l, 

202.  III.  Meaning  of  a  Negative  Exponent. 
If  n  be  an  integer  or  a  fraction,  by  the  Index  Law, 

a^Xa-^-a'^-^-a^^l 

a"""  =  — ,ora"  = ^« 


Ex.  1. 

4-  =  l=i- 

4^      16 

Ex.  2. 

8-4  =  1=1. 

ol     4 

203.  Transference  of  Factors  in  Terms  of  a  Fraction. 

It  follows  from  the  meaning  of  a  negative  exponent  that  any 
factor  may  be  transferred  from  the  numerator  to  the  denominator 
of  a  fraction,  or  vice  versd,  provided  the  sign  of  the  exponent  of 
the  factor  be  changed. 


EXPONENTS.  217 

Ex.   1.  Transfer  to  the  numerator  the  factors  of  the  de- 
nominator of 

xy~^ 

— —  =  a6-'a;-'3/"^i  Remit. 
xy~^ 

Ex.  2.   Transfer  factors  in  the  terms  of ,  so  as  to 

_  2  ' 

xy    ^z  ~ ' 
make  all  exponents  positive. 

2a- 'h        2byh    _      . 
=  -^7—)  Besult 

It  will  not  be  a  difficult  exercise  for  the  student  to  prove 

a** 
Law  II.,  —  =  a'"~",  for  fractional  and  negative  exponents. 

EXERCISE  74. 

Transfer  to  the  numerator  all  factors  of  the  denominator — 

a  a6'  3a 

1-  -T-        4.  -f-.  7.  — -.  10. 


^f 

•J 

'■^- 

xz    ' 

4m-'n"^ 

xpress  wit! 

I  positive  exponents^ 

13.  7x-\ 

16. 

6a -^6'. 

14.  Sab-\ 

17. 

3a"^6-«. 

1 

2-^a;V 
7 

4 

5ia;"^z" 

-i 

11. 


12.        i    _M 

(j!*x     " 


19. 


hd^x 


on       3«"'^ 

1  20 


15.  a'6    ^.  18.  a6-V2/    ^,  cd 


218  ALGEBRA. 

21.  J^.  23.  7'^-'^"*  25 


^  3x-"2/-^ 


«^     '  *  '    '^;>-^.;-^  5-'cZ~n 


Sb-'y 


3„.-  1 


22.  ^^.  24.  ^!^I^.  „„    BZIJ^^ 

Find  the  numerical  value  of — 

27.  4~i  33.^ 39.  (-125)"* 

34.  3-^X44  40.  -tl?^. 


28. 

27"*. 

29. 

^w 

30. 

1 

5-' 

31. 

8-« 

32. 

1/S1-*. 

35.  2-*8-*  '-^> 


36.1-.^  41.  (5J)-^. 


(!)■ 
•(-I)" 


-(s)'0 


--'     -iS^ 


Simplify  the  following  by  performing  the  indicated  opera- 
tions, and  reducing  the  results : 

44.  2a^Xa-\ 


45. 

a'x 

-^Xa- 

^x-'. 

46. 

ba- 

-'X2aV. 

47. 

a*- 

j-a~'. 

48. 

4x- 

-'^2x- 

-3 

49. 

a:2/- 

-'^xY 

-4 

50. 

a*. 

.3a-i 

01, 

C" 

^(#-f-c- 

'A 

52. 

m    "^n-mn'^. 

53. 

6a4a;"*-aV. 

54. 

a"'4.2a^. 

55. 

So;  ~  ^2/  "^  4a;^2/'. 

56. 

4x*  ^  3a:/. 

57. 

7a'a;  ~  ^  -^  bz^y. 

EXPONENTS.  219 

58.  a*l^^.:.*l^^.  Q^^^J^Ziy^. 


59.  X    ^^y^^x%^. 


7nWx 
^  6. — ,  65, 


bU. 


_  1 


x    ^Vx^  x'^Vy 

66. 


61.  —.--—• 


2  «' — :;:i 

xyy 
67.  — ^-^— 


Vx 

62.  a*6"^l^^^-  T^'cl/5^  2/"^^^. 

63.  ,        •  68.   ^        ^ 


3  |/a    ^  a;j/2;^^^ 

204.  IV.  {ary  =  a*""  for  Fractional  and  Negative  Bx^ 
ponents.  It  will  now  be  found  that  using  the  meanings  for 
fractional  and  negative  exponents  which  have  been  deter- 
mined (Arts.  200,  202),  Law  III.,  {ary  =  a""»  applies  to  theni 
also. 

First,  when  n  is  a  positive  fraction,  -,  the  terms  of  the 

fraction,  p  and  q,  being  positive  integers. 


l{a-yr- 

=  {ary 

Extracting 

the  g'*  root  of  both  sides, 

p 
{a'^y  - 

-a* 

Substitute 

tifor^,      {a'^y- 
9. 

=  «»»". 

Second,  when  n  is  a  negative  integer  or  negative  fraction ; 
as,  —  t. 


220  ALGEBRA. 

rarY  =  (dr)-*  =  — = —  =  —  ^a-^'^a"^. 

^        («*")  +  '     or* 

It  will  not  be  a  difficult  exercise  for  the  student  to  provn 
also  Law  V.,  {ahy  =  oJ'h''. 

First,  when  n  is  a  positive  fraction. 
Second,  when  n  is  a  negative  quantity. 

-3    -4 

Ex.  1.  Find  the  value  of  (4    2)    ^ 

{^-i)~^  =  4.^  =  Z%  Result. 
Ex.  2.  ^[^a'^y  =  (8a"^)*  =  8*a"* - 4'  ^^^^' 


Ex.3.   /lg^V*=    ^^"^^'  ^gl!^ 

\  «i^'  /       81-^6-4       let 

27a^6^     ^     ,, 
=  — - — )  Uemlt. 
8 

EXERCISE   75. 

Reduce  to  the  simplest  form — 


1.  {a')-\  7.  {a'b~^)~\  13.  (5a;~i)"* 

2.  (a; -=*)"*•  8.  {x~^y^)~\  14.  (8a') "■^. 

3.  (a-')*.  9.  (8-^*  15.  (4a; -*)"*. 

4.  (x~*)*.  10.  (64-0"*.  16.  (9a-'a;^2/~')~*. 
6.  (c^)~^  11.  (9^)  ~*.  17.  (- 2a'a;~  V. 
6.  (a~*)"*  12.  (3a-%  IS.  i~-bx-'y^-\ 


EXPONENTS. 


221 


19.  (9x-'3/-»)"f 

20.  (a'Vo^"'. 

21-  (aVa^~^. 


22.  l^(a^^r^. 

23.  (aVS^^^"'. 


26.  (c'-a;-'-) 


27 


l2bVx\   * 


28 


1  82/V^~ 
29.   {.^p^ 


l/a^yj 


l-i-pf 


30. 


31. 


l>'8a-»6V? 


32.  l/x-^V't^^v'yi/^^. 

33.  l^a6-^c-^X  V'SP?: 

34.  (xV^^"'Xi/5:"^^. 

36.  f-^^xi^f- 
38.  VlV(H)"*]. 
39.   [\(-U)"*]      • 


41.  8    ^  +  9^-2"   +1 


7». 


42. 


_  1 


Va^     V7^     b    ^i^ 


l^P^ 


-0 


43.  fa%~^»/a-=^5'T/?j. 


222 


ALGEBRA. 


45. 


laJx-' 


X 


/  aVx 
i 


46. 


47. 


m'<stHM 


y-^ 


Ex.  Expand  (a:^  -  4a;  "^) 

=  (J/  -  3(a;V(4a:~^)  +  3(a;^)  (4a;"  V  -  (4a;  "^)' 
=  x^  -  \2x^  +  48a;  ~  ^  -  64x  "  ^,  iJesw^i. 


Expand — 

48.  (2a; -3a; -7. 

49.  (l/^-2v'i)*. 

60.  (3v^+2Vx^^ 

61.  (ix"*-2a;4).* 


52.(21^  +  ^)'- 


/        2VaF^Y 


Ex.  Solve  a;    ^  =  27.    Kaise  both  sides  to  the  power  ( -  |). 
(a;-^)     ^  =  (27)-^  =  --  =  i        .'.x  =  \. 


27^ 


Find  the  value  of  x  in  each  of  the  following — 

55.  a;^  =  2.  58.  a;"^  =  4.  61.  a;""  =  2. 

3 


8  _  3  —1 

66.  «^  =  _  27.  69.  a;    ^  =  —  1.  62.  x    «  =  —  3. 


67.  X 


4^3. 


60.  X 


1, 


63.  a; 


-i  = 


=  -jV. 


EXPONENTS. 


223 


205.  Polynomials  whose  Terms  contain  Fractional  or 

Negative  Exponents. 

Ex.  1.   Multiply  x  +  Zx^  —  2x^  by  3  -  2x~^  -h 4a;~^. 


2a;^ 


x  +  3a;^ 
3-2a;~^  +  4a;"^ 
Zx  +  9a;^     -  6a;^ 
-  2a;^     -  6a;^  +  4 


+  4a;3  +  12  -  8a; 


-\ 


Zx  +  1x^     -  8a;^  +  16  -  8a; "  ^,  Product 
Ex.  2.  Extract  the  square  root  of 


Vx 


4 

•  -  4-  25x 
a; 


* 


241 


16 

+  7' 


writing  the  given  expression  by  use  of  exponents  only. 

1  +  4a;"^-  2x~^  -  4x~^  +  25a;~^-  24a;"'^  +  16a;"* 
1  11 +  2a;~^-3a;~^  +  4a;"* 


2+2a;"^ 

4x~^  -2x~^ 
4x~^  +  4x~^ 

2  +  4a;~^-; 

3a;"^ 

-6.-t- 
-6a;-t- 

4a;"'  +  25a;"^ 
12x'~*+    9x~^ 

2  +  4a;~*-6a; 

~^  +  4a;"' 

Sx~^  +  16a;~3  - 
8a;~'+16a;~^- 

-24a;~^  +  16a;  ~* 
-24a;"^  +  16a;"' 

I 


206.  Summary  of  Principles  Relating  to  Exponents. 

1.  In  a  fractional  exponent  the  numerator  denotes  a  power y  the 

23  ^ 

denominator  a  root.    Exs.  a^  =  Va^ ;  32"^  =  8. 

2.  a'=  1.  5°  =  1. 


3.  a 


224  ALGEBRA. 

4.  In  the  use  of  exponents,  fractional  and  negative  as  well  as 
positive^  use  the  rules  which  govern  the  use  of  positive  integral 
exponents  ; 

That  is,  in  brief, 

(1)  To  multiply,  add  the  exponents. 

(2)  To  divide,  subtract  the  exponent  of  the  divisor  from  the 

exponent  of  the  dividend. 

(3)  To  raise  to  a  power,  multiply  the  exponents. 

(4)  To  extract  a  root,  divide  the  exponent  by  the  index  of  the 

root. 

,   ,  .    ,  EXERCISE   76. 

Multiply — 

1.  a-2a^  +  3by  2a*  +  3. 

2.  a*-a*  +  l  by  a^  +  1.  ' 

3.  Zx^  —  2x^yi  +  32/*  by  Sx^  +  2y^. 

4.  2a;*-3xi"  +  4by2  +  3a;"i 

6.  a-^-a~h^-\-bhy  a-^  +  a~h^  +  b. 

6.  x'-xy  +  2y^hy2x-'  +  x-'y-''  +  y-\ 

7.  2x^-Sy-^  +  x~^y-^hy2x~^y  +  Sx~\ 

8.  2x^-Sxi-A-\-x~^hyZx^+x-2xi. 

9.  a^x~i  +  2  +  a~hihy  2a~  %^  — 4a"%^  +  2a-'A 


1     _     41^1? 
10.  2Vx  +  3x^1^2/  +  —^  ^y 


3  3     ^21^^ 


^   '  vx   iyxif    ^ 

Divide — 

11.  5x  +  2x*  —  2x^  +  1  by  x^  +  1. 

12.  Sx-''+  —  -{-Sy-'-lSxy-'-8x'y-*  by  2x-^-f3y-* 

xy        "^ 

4-  ixy  ~  * 


EXPONENTS.  225 

13.  a;"*-a:"*  +  5-2.'c*by  1+2VX. 

14.  V^—PyhyVx.—  V'y. 

15.  Vd+V'^^  1/5  by  Va+  ]^d5  +  i^E. 

16.  27a=^  -  SOay  -'  +  Sy-^hySa-  2aJy  ~^-y-\ 

17.  x~^  +  x~%-^  +  y-'hy  x-'  +  x'^y-"^  +  x'^y-\ 

or  2         1  o'?/  1 

18.  --x^2/"^-4.l^--ir^by  1^  +  21/^. 

9      3V^      lOx       1/^  3 

«       l/a'       Va        Fa  l^a 

20.  4l^^-8l>'a-5  +  4?=  +  -|-by2aA-t^ 1.. 

Va      V(r  Va 

Extract  the  square  root  of — 

21,  x^—^x^y^ ^-A.xy.  22.  9x1/"'  + 12y-*  +  4»-*. 

23.  a-^'-Wh^  +  10h~12ah^  +  ^ah\  '. 

24.  a;"*  +  8a;-^-2a;"^+16a;"^-8x'i  +  l. 

25.  92;  - '  -  30:c  "  %  +  1 3x  -  y  +  20x  "  V  +  4a; "  y. 

26.  25a*6-»-10a%"^-49  +  10a"*6*  +  25a"^6\ 


2^^^_18v^^l%_6iy  ^ 
'  a;*  a;'  x^  x 

^  ,      241/?        4x^         ^    /~i  4 

28.  9ar^  -  — ,— -  +  -r^  +  16^—  +  — !;=• 

y^        y  2S  2  9 

^^*  4a;      3^1      9]/^      y^      V" 


CHAPTER    XVII. 
RADICALS. 

207.  Indicated  Roots.  The  root  of  a  quantity  may  be 
indicated  in  two  ways : 

(1)  By  the  use  of  a  fractional  exponent;  as  a^. 

(2)  By  the  use  of  a  radical  sign ;  as  Va. 

For  some  purposes,  one  of  these  methods  is  better;  for  some, 
the  other. 

2  1 

Thus,  when  we  have  a^  Xa^  Xa~^,  where  the  quantities 
are  alike  except  in  their  exponents,  it  is  better  to  use  frac- 
tional exponents  to  indicate  roots ;  but  if  we  have  5VZ 
—  7l/27  +  8l/12,  where  exponents  are  alike,  but  coefficients 
and  bases  unlike,  it  is  better  to  use  the  radical  sign  to  indi- 
cate roots. 

In  the  preceding  chapter  we  considered  exponents ;  we 
have  now  to  investigate  the  properties  of  radicals. 

208.  A  Radical  is  a  root  of  a  quantity  indicated  by  the 
use  of  the  radical  sign.     Exs.  Vx,   VT7. 

209.  Surds.  An  indicated  root  which  may  be  exactly 
extracted  is  said  to  be  Rational.  Ex.  1/27,  since  the  cube 
root  of  27  is  3. 

An  indicated  root  which  cannot  be  exactly  extracted  is 
called  a  Surd.     Exs.  VB,  v6. 

210.  The  CoeflBcient  of  a  radical  is  the  number  prefixed 
to  the  radical  proper,  to  show  how  many  times  the  radical  is 
taken. 

Ex.   The  coefficient  of  5l/3  is  5;  of  Qav'x  is  6a. 

226 


RADICALS.  227 

211.  Entire  Surds.  If  a  surd  have  unity  for  its  coefficient, 
it  is  said  to  be  Entire. 

212.  The  Degree  of  a  radical  is  the  number  of  the  indicated 
root.     Ex.  Vx  is  a  radical  of  the  third  degree. 

213.  Similar  Radicals  are  those  which  have  the  same 
quantity  under  the  radical  sign  and  the  same  index.  (The 
coefficients  and  signs  of  the  radicals  may  be  unlike ;  hence, 
similar  radicals  must  be  alike  in  two  respects,  and  may  be 
unlike  in  two  other  respects.)  Ex.  .5t/3,  —  4v^  are  similar 
radicals. 

214.  Fundamental  Principle.  Since  a  radical  and  a 
quantity  affected  by  a  fractional  exponent  differ  only  in 
form,  in  investigating  the  properties  of  radicals  we  may 
use  all  the  principles  demonstrated  concerning  fractional 
exponents. 

Thus,  since  (aby  =  a'^i**  is  true,  when  n  is  a  fraction,  as  i, 


(aby  =a^b^ 

.  • .  i/'oS  =  i^a  •  y^. 

TRANSFORMATIONS  OP  RADICALS. 

215.  I.  Simplification  of  a  Quantity  under  Radical  Sigm. 

If  a  factor  of  the  quantity  under  the  radical  sign  is  a  perfect 
power  of  the  same  degree  with  the  radical,  the  root  of  this 
factor  may  be  extracted  and  set  outside  as  a  factor  of  the 
coefficient. 
Ex.  1   Simplify  1^56. 

1^=^WX7  =2^7,  Result    (Art.  214.) 

Ex.2.   Simplify  6  l/18a'6V. 

5 l/18a'6V  =  5 VWWX^M=  15abc' V^Ec,  Remit. 

Hence,  in  general, 

Separate  the  quantity  under  the  radical  sign  into  two  factors^  one 


228  ALGEBRA. 

qf  which  is  the  greatest  perfect  power  of  the  same  degree  as  the  rad* 
ical ; 

Extract  the  required  root  of  this  factor^  and  multiply  the  coefficien' 
of  the  radical  by  the  result ; 

The  other  factor  remains  under  the  radical  sign. 

216.  Quantity  under  Radical  Sign  a  Fraction.  To  sim- 
plify in  this  case, 

Multiply  both  numerator  and  denominator  of  the  fraction  by  such 
a  quantity  as  will  make  the-  denominator  a  perfect  power  of  the  same 
degree  as  the  radical; 

Proceed  as  in  Art.  215. 

Ex.  1.   Simplify  ]^^. 
1^^  =  v'^fXi  =  V'W  =  l^ift^Xl5  =  J l^,  Remit, 

Ex.2.   Simplify  ^-^^ 


/3^_    j'hax''       2b  ^    fWiM 


-^/ 


— 2 

—  X  lOab  =  —VWab,  Result. 
366'  6& 


217.  Meaning  of  Simplification.  By  simplication  radicals 
are  reduced  to  their  prime  form,  so  that  it  is  made  easier  to 
determine,  for  instance,  whether  a  number  of  given  radicals 
are  similar  or  not. 

Thus,  it  is  difficult  to  say  whether  71^18,  —5V12  are  sim- 
ilar, but  when  the  given  radicals  are  put  in  the  form  21  V2, 
—  30 1/2,  it  is  easy  to  see  that  they  are  similar. 

Again,  the  radicals  (a  — l)-v/ and  (a  +  1) -y — -— , 

although  unlike  in  present  form,  may  be  reduced  not  only  to 
similar  radicals,  but  to  the  same  expression,  Va^  —  1. 
The  pupil  should  show  this  reduction  for  himself. 


MABICALS.  229 

EXERCISE   77. 

Express  in  the  simplest  form — 

1.  1/12.  11.  ^2i.  21.  1/200^. 

2.  1/18.  12.  ;^^54.  22.  |/Ii7^. 

3.  1/27.  13.  1,^72.  23.  -2i/^63?y^ 

4.  -1/2D.  14.  -f|/108.  24.  v'  -81aV. 

5.  21/24.  15.  |>'i8.  25.  i/aXx-y)'. 

6.  -31/28.  16.  v*^28^^.  26.  |/49ar'(a  +  1/. 

7.  il/4i.  17.    1^250^^.  pro-r;^ 

27.  10,  12acn^ 

8.  J 1/45.  18.  |/99a.  >/    25x* 

9.  11/50.  19.  2^/4^^.  28.  3    /n2i^ 


mV^ 


10.  i^^iB.  20.  rt|/8^^  ^     9a» 

Simplify — 

^-   ^-  36.  f#.  *2-  ^l^'?- 

^3a 


30.  2i/|.  "-  /45?5? 

31.  3V^.  St'KI^  ''-'VsxV' 

38.  31^.  "*•  "V^- 
33.   VH. 

39.  5a^.  |I2(^^, 
34-  V^-                  40.  -3i^|.                       ^5(x  +  y) 

-WIf-       '■•-'=<S-    --V^' 


"■  <"  +  »>\'5^^-         "■'-^ 


230  ALGEBRA. 

EXERCISE    78. 

ORAL. 

Reduce  by  inspection — 


1.  VE. 

6.  VI 

11.  ^7t- 

16.  ^^' 

2.  Va^. 

7.  n. 

''■  €• 

17.  21/41-. 

3.  V'a^xK 

8.  VI 

13.  3l/f. 

18.  |l/12i. 

4.  v'le^^. 

9.JI- 
10.  l^f. 

14.  2Vf. 
IK       '« 

19.  l/2f. 

5.  V27xY- 

20.  11/3^. 

218.  Making  Entire  Surds.  It  is  sometimes  desired  to 
introduce  the  coefficient  of  a  radical  under  the  radical  sign. 
This  may  be  done  by  simply  reversing  the  process  of  Art. 
215. 

Ex.  1.    Express  3  {^5  as  an  entire  surd. 

3i:^  =  V^^'X5--=--l>'T35. 

Ex.  2.  - 2l>^  =  -  1^96  =  1^=^. 

Ex.3.  -2v*3  =  -|y48. 


EXERCISE   79. 

Express  as  entire  surds — 

1.  2T/S.  7.  2v^  3.. 

2-  3 1/5.  g^  2w.l/3m: 

4.  -21/5.  ^ 


6.  -3i>^.  10.  i>/6^ 

6.  -2v'^^=^         11.  fi/io: 


12. 

3m  J  271 
491  ^9m'^ 

13. 

.,fc». 

14. 

-^'^- 

15.  (x-l)V^. 

17.  "-Vs"— 2?: 

a  +  b 

-■<'"'><iZ- 

18.  (l-x)J-^ 
*  x  —  1 

231 


219.  II.  Simplification  of  Indices.  If  the  exponent  of 
the  quantity  under  the  radical  sign  and  the  index  of  the 
radical  sign  have  a  common  factor,  this  factor  may  be  can- 
celed and  the  radical  thereby  simplified. 

Ex.1.   ^E'  =  J  =  J  =  ^. 

Ex.  2.  v"^125=|^'5'  =  |/5. 

EXERCISE   80. 

Simplify  the  indices — 

1.  i>'a\  6.  V^z-^TO.  9.  {TS^aVy. 

2.  ]^'^.  6.  v^lOD^^.  10.  v^9?^ 

3.  1^  o^.  7.  ^gaW^.  11.  p^^'y'^. 

4.  ^W.  8.  i^El^P.  12.  6|?  2|. 

220.  III.  Reducing  Radicals  to  the  Same  Index.  Radi- 
cals of  different  degrees  may  be  reduced  to  equivalent  radicals 
of  the  same  degree. 

Ex.  1.  Reduce  |/2  and  y'S  to  equivalent  radicals  having 
the    same  index. 

v/2  =  2*  =  2*  =  ir2'  =  i?'"B 

Ex.  2.  Arrange  in  ascending  order  of  magnitude  |^5,  y^Wy 
V2. 

We  obtain  Y/T2^,  ^'81,  y'M ; 

hence,  the  ascending  order  of  magnitude  is,  ]/2j  ^^3,  |^5. 


232  ALGEBRA. 

EXERCISE   81. 

Reduce  to  equivalent  radicals  of  the  same  (lowest)  degree — 

1.  VI'sLXid  ^U.  7.  ^%  1^9,  ^5. 

2.  V  5  and  ^%  8.  Vd,  ^/aF,  ^oF. 

3.  |>  B  and  ^'5'  9.  ^Ta,  ^2b,  ^^^. 


4.  i/f  and  y^.  10.  yx-{-y  and  y  x  —  y. 

6.  1^1^  and  i^25.  H-  y^  and  v^^. 

6.  T/6"and  1^200.  12.  |/V^,  yV,  ]^cf. 

Which  is  greater— 

13.  l/3~or  i^T?  17.  i^  10  or  2i^f? 

14.  1^15  or  1/6  ?  18.  T/2|orv^4j? 

15.  V^or  I'/n  ?  19.  3v'  6  or  2T/5f  ? 

16.  v>^23  or  2^/2?  20.  l/f  or  v'^S? 

Which  is  the  greatest — 
21.  1/B;  1^5;  or  i]^^^?  22.  S^^^T^,  2l/6^  2i^li}? 

OPERATIONS  WITH  RADICALS. 
I.  Addition  and  Subtraction  of  Radicals. 

221.  The  Addition  of  Similar  Radicals  is  performed  like 
the  addition  of  similar  terms,  by  taking  the  algebraic  sum  of 
their  coefficients. 

The  Addition  of  Dissimilar  Radicals  can  only  be  indicated. 

Ex.  1.   Add  Vim  -  2 1/50  +  1/72  -  1/18. 

1/I2B -2V50  +  1/72 -  l/IS  =  8i/2 -  10i/2  +  6l/2 - 3l/2 

=  1/2,  Sum, 


RADICALS.  233 

Ex.  2.  2l/|  +  il/6D  +  1/15  +  i/| 

=  I  Vl'5  +  i  Vio  +  1/15  +  i  1/15 

Ex.3.  vi28  +  2|^  r--3i.'/SI 

=5j^2"-V3',  >^m. 

EXERCISE   82. 

Collect — 

1.  /IB  + 1/8.  9.  2i>^lB9  -  iMiB. 

2.  1/50  -  1/32.  10.  1^  24  +  v''  81  -  iXBTS. 

3.  21/27  +  1/75.  11.  VI  +  1/f. 

4.  31/90-51/40.  12.  2l/f  4-  1/48. 
6.  1/5  +  1/2D  +  1/45. 


13. 


%^4i 


6.  4|^1B-2|^54. 

7.  3i>H25- 4^^/135.  14.  i^^^  +  ii^l^. 

8.  1^^162  +  3|;V48.  15.  |v>^  j  -  f  i^f 
16.21/25^+31/4^-21/365^. 

17.  3|>  2c  +  3^^' 54^  -  |^'2000c. 

18.  1/12^ +  6 1/48"^ -6 1/3^^ 

19.  2l/^"W-  3a  1/166?  +  5cl/9^?5: 

20.  h^Ta  +  1^^  250^  -  26i>132^. 

21.  1/2  +  1/18  -  1/5D  +  1/IB2. 

22.  1/73 -41/^3 +  2V108. 

23.  6l/f-5l/24  +  12l/f: 

24.  5  Vf-  12l/f  +  61/60  -  30 1/^. 

25.  3 1/5- 10  V"f  +  2 1/45  -  5  V^. 

26.  1/27  -  1/18 +1/3D0-1/1B2'+ 6 1/2-7 1/3; 


234  ALGEBRA. 

27.  21/63  -  Syf  -  l/f  +  il/4o  -  4 1/7: 

28.  l/2iB -\-V4E-  1/768 4- 9l/f  + 1/75 - 3 i/'33f. 

29.  21  >/f -  5 1/4  +  6 1/4J  - 10  l/3i  +  ^0-  Vn\. 

30.  5a  l/r2W^  -  36  l/27a'  +  2  VHOO^OT  -  40a6  >/p. 

n.  Multiplication  of  Radicals.       *    ,, 
222.  Multiplication  of  Monomials. 
Since  by  the  commutative  law, 

=  ac  vbdj 
we  have  the  general  rule, 

Reduce  the  radicals  if  necessary  to  the  same  index; 
Multiply  the  coefficients  together  for  a  new  coefficient ; 
Multiply  the  quantities  under  the  radical  sign  together  for  a  new 
quantity  under  the  radical  sign; 
Simplify  the  result. 

Ex.  1.  Multiply  Sl/e'by  2VW. 

5  VWX  2  \/W=  10  |/I8  =  30 1/2;  Product. 

Ex.  2.  5|/JX  2i/'/3"=  5],«/8  X  2^9 

=  101^72,  Product. 


4 


t7^,  Product. 


2 

223.  Multiplication  of  Polynomials.  The  Distributive 
Law  applies  here  as  in  ordinary  algebraic  multiplication  of 
polynomials;  hence, 

Reduce  each  term  of  the  mxdtiplier  and  multiplicand  to  its  stm- 
plest  form; 


BADWALS.  236 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the 
multiplier  ; 

Simplify  each  term  of  the  result,  and  collect. 

Ex.  1.   Multiply  31/2'+  5 1/3"  by  3l/2'-  1/37 
31/2  +  51/3 
31/2  -  1/3 


18  +  ISl/S" 

-  31/6  -  15 

3  +  121/6,  Prodvxit. 

Multiply  V6  -  21/12  +  5  T/S"by  3  l/T-  5 

JV^ 

&  -  21/12  +  51/3  =  1/6  -  4V3  +  5V'3  = 

=  1/3  + 

1/5 

1/3  +  1/^ 

31/3  -  21/2 

9  +  31/18  -  21/6  -  21/1^ 
=.9  +  91/^  -  21/^  -  41/3,  ProducL 

EXERCISE  83. 

Multiply — 

1.  l/3^by  21/12.  11.  l/a  by  ^a?F. 

2.  3 1/5" by  1/13.  12.  l/2  by  |^3: 

3.  2v'  4  by  31^6.  13.  |^2^^by  V^, 
4. 1^24  by  v''4.  14.  v  9  by  t/ST 

6.  3 1/18  by  2 1/I2.  15.  i^^F  by  ^/A. 

6.  21^15  by  31^35.  16.  vTS  by  l/l^. 

7.  1 1/28  by  f  1/35.  '  17.  l^f  by  v'^. 

8.  il/fby|l/ff.  .  18.  l*/f  by  l^V. 


9.  4n_byiin.  19    AW^AWl. 

10.  f  v''' A  by  ||7f .  *    ^Z  3x-^^      A/  4a6* 


'3S6  ALGEBRA, 

20.  V'SX  Vji.  23.  l^^X  V^. 

21.  V^SX  VT  24.  V^ff  X  V'^X  I^. 

22.  I^HI  X  V- W-  25.  V'^  X  V^H  X  V^360. 

26.  VS-  V'6  +  2  l/ID  by  2  ^27 

27.  3l/5~-  t/IO  4-  2l/r5  by  4l/5: 

28.  4>/6~-3v/3>  31/2" by  2l/6: 

29.  i  1/f  +  i  Vj-  f  Vjby  20  VT2. 

30.  10 1/f -  5  Vf+  14 1/ff  by  I V^. 

31.  3+  1/2  by  2-21/2"  32.  5  -  2V^  hj  A  i- BVK 

33.  2  VT-  3 1/2^  by  4 1/3  +  5 1/2: 

34.  3  VW+  5 1/5^  by  5 1/3^-  3  VK 

35.  4 1/2"- 3  l/3^by  3 1/2~+ 4  l/S: 

36.  3  V5-  2  V2+  1/3  by  3 1/5"-  VW. 

37.  1/2"+  1/3"-  1/5"  by  V2-  VW+  VK 

38.  2 1/3"- 3 1/6 -4 1/15  by  2 1/3  +  31/6^+4 1/15. 

39.  3  l/eO  +  2  V5'-  3  l/6"by  2 1/5"+  3  VW. 

40.  il/8"+  1/S2  -  1/18  by  SVS-iVS2  +  2l/l^. 

41.  12l^-4l/f+4l/2l6by  6l/f-2l/f +  31/6: 

42.  V2x  +  1/^=1  by  VEx. 

43.  l/Bx  +  1/^+T  by  y^TT. 

44.  l/x^^  -  3l/^Tl  by  2VxT\. 

45.  a  —  Va  —  x-\-  l/oTby  1/a  —  x  +  l/o: 

46.  31/2^-51/^^=1  by  31/2^  +  51/^^^1. 

47.  VaTx  —  Va  —  x  by  Va-\-x  +  l/a  — a;. 

48.  (21/2"+  1/S")  (31/^-  l/S")  (31/3""-  l/2). 

49.  (2 1/^+2  +  3 1/2)  (6a; - 5 VWV4.)  (3 l/F+2 - 2 V^. 


RADICALS.  2<J7 

in.  Division  of  Radicals. 

224.  Division  of  Monomials.     Reversing  the  process  for 

multiplication,  we  have  the  rule, 

//  necessary,  reduce  the  radicals  to  the  same  index ; 

Find  the  quotient  of  the  coefficients  for  a  new  coefficient,  and  the 
quotient  of  the  quantities  under  the  radical  signs  for  a  new  quarUity 
under  the  radical ; 

Simplify  the  result. 

Ex.  1.   Divide  Gl/S'by  Sl/BT 
6V/8 


2l/f  =  21/9  =  11/3;  Quotient, 


81/6 
Ex.  2.    Divide  Gi^S'by  2i/2. 

2^^  2^>¥^'^'2^X2^^'W  =  *^^^'  ^''''^' 

EXERCISE   84. 

Divide — 

1.  VT  by  VB.  11.  4l/|  by  W^.       • 

2.  41/12  by  21/6.  12.  5l4"|  by  21/^. 

3.  121/15  by  41/5.  13.  f  l/JI  by  ^V^f. 

4.  21/60  by  3l/5.  14.  21^3  by  3l/2. 
6.  81/125  by  lOl/lO.  15.  |/55  by  v*  36. 

6.  31/405  by  9l/45.  16.  |>  12  by  ^^. 

7.  a»  1/^5^" by  2al/a'5:  17.  ^"^  by  v'^^S- 

8.  41/18  by  51/82.  18.   I/6J  by  x''^' 

9.  3 1/iD  by  5 1/28.  19.  3  l/ff  by  2  l^ff |. 
10.   l/ifby  >/f.  20.  i^W  by  2i^~ft. 


238  ALGEBRA, 

21.  5 1/35 --7 1/20  by  1/5. 

22.  31/6 +  91/3  by  31/1 

23.  12l/7-60V5by4VS. 

24.  6vlD5  +  18l/40-45l/12by3l/m 

25.  8  >/45  -  15 1/24  -  1/60  by  2  V/30. 

26.  12  v'lB  +  30 1^2T)  +  42  y  30  by  2  K'lB. 

27.  10v'i8~4l^60  +  5l^l00by  3v'3D. 

225.  Rationalizing  a  Monomial  Denominator.  If  the 
denominator  of  a  fraction  be  a  surd,  in  order  to  make  the 
denominator  rational, 

Multiply  both  numerator  and  denominator  by  such  a  number  as 
will  make  the  denominator  rational. 

5         5         1^4      51^ 

^"7i  =  72^7^^W=^^^'^''"^'- 

One  object  in  thus  rationalizing  the  denominator  of  a  frac- 
tion is  to  diminish  the  labor  of  finding  the  approximate 
value  of  the  fraction.      Thus,  if  we  find  the  approximate 

5 
numerical  value  of  — -  directly,  we  must  find  the  cube  root 

of  2,  and  divide  5  by  the  decimal  which  we  obtain.  On  the 
other  hand,  if  we  find  the  value  of  the  equivalent  expression, 
|v''4,  we  extract  the  cube  root  of  4,  multiply  by  5,  and  divide 
by  2.  In  the  latter  process  we  therefore  avoid  the  tedious 
long  division,  and  diminish  the  labor  of  the  process  by  nearly 
one-half. 

226.  Rationalizing  a  Binomial  or  Trinomial  Denomi- 
nator. If  the  denominator  of  a  fraction  be  a  binomial  con- 
taining radicals  of  the  second  degree,  since 

(l/a4-  l^b)(V^-Vb)  =  a-b, 


RADICALS.  239 

Multiply  both  numerator  and  denominator  by  the  denominator^ 
with  one  of  its  signs  changed; 

For  a  trinomial  denominator  repeat  the  process. 

21/5  +  41/3       21/5  +  41/B       SVB  +  V^ 
31/5 -v^B        31/5 -i/B       31/5+1/3 
42  +  141/15      42  +  141/15      3+1/15 
="      45-3      -         42         =        3       '^''^^^• 

^^  4  4  1+1/3-1/2 

'    '  1  +  1/3  +  1/2      1+1/3+1/2      1+1/3-1/2 

2(1  +  1/3  -  1/2)      1-1/3     ^       ,^        /n  ^     , 

=  -^ '  X =2+1/2-1/6,  Result 

1+1/3  1-1/3 

EXERCISE  85. 

Reduce  to  equivalent  fractions  having  rational  denomi- 
nators— 

1  1-1/2  1^-1 

1. 4. 7.    '^     3_    • 

V2  1/6  31^2 

1/2  2+1/5  2v^^-3i?^i 

^'21^3*  ^*~2l^*  *         5i?6 

2  31/2-1/3  3-1/2 

3. 6. 9. -• 

3i/5  2l/6  3+1/2 

5+1/3  21/13 +  31/ID 

^^*  2^v¥*  "^^*    41/3  +  31^2 

31/3-21/2  SVa-4VB 

11.  ^ .  14.  = 

21/3  +  31/2  2l/a-3i/5 

31/6-21/3  1/^1^1  +  3 

12. 15.       

4l/6~3l/B  l/a;  +  l  +  2 


21/^  +  31/2+1/42 
Vx'  -  1  -  l/x'  +  1 

Vx'  -  1  +  1/x"^  +  1 
Va+  Vb—Va  +  b 

240  ALGEBRA, 


2V2a-l  +  ZVa  21/^-31/2-1/52 

16.    =:^ 19. 

3l/2a  — l  +  2l/a 

2  +  1/B  -  1/2 

17. — .  20. 

2-VQ+  1/2 

1/5-1/6+1 

18.  — z = 21.  _  _ 

1/6+1/5  +  1  Va-Vb+VaTh 

Find  the  approximate  numerical  value  of— 

3                                 1  1/3-1/2 

22. 25. 28. 


1/2  1/300  1/2+1^ 

21/5  31/7  51/7-1 

23.  — -.  26. 29. 

31/2  51/5  1/7  +  2 

12  1  +  1/2  31/5-4 

24. 27. 30 


VI  2-1/5"  41/3-5 

IV.  Involution  and  Evolution  op  Radicals. 
227.  The  process  of  raising  a  radical  to  a  powei,  or  extract- 
ing a  required  root  of  a  radical,  is  usually  performed  most 
readily  by  the  use  of  fractional  exponents. 

Ex.  1.   Find  the  square  of  3l^£ 

(3  V'x)'  =  (Sx^y  =  dx^-  =  9  l^i». 
Ex.2.   Extract  the  square  root  of  4a  l/o^. 
(4a  1/^6^*  =  (4a  •  ah^)i==  4*a%^6* 

=  2a^6*  =  2  lV&^  =  2a6  v'^,  Result. 
Ex.  3.   Extract  the  cube  root  of  l^o^. 

(V¥F)^  =  (a'b^)i  =  ab^  =  aV¥  =  abVl. 
This  process  might  have  been  performed  by  extracting  the 
cube  root  of  a%^  as  it  stands  under  the  radical  sign ;  thus, 

j/^y^^^  V^j^  =  ab  1/6,  RemiL 


RADICALS.  241 

EXERCISE   86. 

Perform  the  operations  indicated — 

1.  (l^m)*.  5.  (1^^*.  9.  (1^^«. 


2.  (I/?/.  6.  {V^y\  10.  V/64aV8?. 


3.  (1/(7?)'.  7.  (v/2?/.  11.  (V-  l^)«. 


4.  (VSB)'.  8.  (}/~^\  12.  ^F^47M?^. 

V.  Square  Root  of  a  Binomial  Surd. 

228.  A  Quadratic  Surd  is  a  surd  of  the  second  degree. 

Exs.  1/3;  V^. 

A  Binomial  Surd  is  a  binomial  expression,  at  least  one 
term  of  which  contains  a  surd. 

Exs.  V2  +  bVW, 
or  *      a  +  1/6. 

229.  A.  The  product  of  two  dissimilar  quadratic  surds  is  a 
quadratic  surd.     Thus, 

l/2~XT/e'=T/l2-2l/3; 
or  VoB  X  Vabc  =  ah  Vc. 

Proof.  If  the  surds  are  dissimilar,  one  of  them  must  have 
under  the  radical  sign  a  factor  which  the  other  has  not.  This 
factor  must  remain  under  the  radical  sign  in  the  product. 

230.  B.  The  sum  or  difference  of  two  dissimilar  quadratic 
surds  cannot  equal  a  rational  quantity. 

Proof.     If  Vadz  VI  can  equal  a  rational  quantity,  c, 
squaring,  a  ±  2  Vab  -\-b  =  (f, 

±  2  Va6  =  (^  —  a  —  b. 

But  Va6  is  a  surd  by  Art.  229 ;  hence  we  have  a  surd 
equal  to  a  rational  quantity,  which  is  impossible. 

16 


^42.  ALGEBRA. 

231.  C.  If  a+  Vb=x-\-  Vy,  then  a=^x,  b  =  y. 
Proof.     If  a  +  ]/6  =a:  +  l/^; 

transposing,  V^  —  Vy  =  x  —  a. 

If  b  does  not  equal  y,  we  have  the  difference  of  two  surd'" 
equal  to  a  rational  quantity,  which  is  impossible ;  hence, 
b  =  y,        a~x. 
In  like  manner  let  the  student  show  that  if 

a  —  Vb  =  x—  Vy,        then  a  —  x,        b  —  y. 

232.  Extraction   of  Square  Root  of  a  Binomial   Surd. 

If  we  expand  (VW+VSy,  we  obtain  2  +  2l/6~+3,  or  5 
H-  2VW.  Hence,  the  square  root  of  the  binomial  surd 
5  +  2VW  is  1/2 -h  V^.  Hence,  the  square  root  of  some  bi- 
nomial surds  may  be  extracted.  To  investigate  a  method  of 
doing  this,  let  a  +  Vb  be  a  binomial  surd,  and  let  Vx  -f  Vy 
be  its  square  root. 

•'■  Va  +  Vb  =  Vx  +  Vy (1) 

Square  both  sides, 

a  +  1/15  =  x  +  IV  xy  +  y (2) 

.' .  a  =  x^  y,        Vh  =  2Vxy.     (See  Art.  231.) , 

Hence,         a-Vb  =  x  +  y-  2Vxy (3) 

Extract  square  root  of  (3), 

Va  -VB  ^Vx  -Vy (4) 

Multiply  (1)  by  (4), 

Va"  -b  =  x-y (5) 

But  a  =  x^  y (6) 

Adding  (5)  and  (6)  and  dividing  by  2, 
^  _  a  +  Vd}  -  6 
^  2 

Subtracting  (6)  from  (5)  and  dividing  by  2, 

^  2 

2  2 


RADICALS.  243 

Hence,  the  square  root  of  a  binomial  surd  may  always  be 
extracted  in  form,  but  we  get  a  result  simpler  than  the  orig- 
inal one  only  when  d^  —  6  is  a  perfect  square. 

Ex.  Extract  the  square  root  of  5  +  2VK 


Let 

Vx  +  V^  =  V  5  +  21/6 

Then 

Vx-Vy  =  Vb-  21/6. 

{See  (4)  above.} 

Multiply! 

Qg,             a; -y  =  1/25 -24 
.-.  x-2/  =  1 

But. 

x  +  y  =  5 
,'.  x  =  3 

.-.    Vx  +  Vy  =  1/3  +  1/2 

Vb  +  21/6"  =  1/3  +  1/2,  The  required  root 

233.  Square  Root  of  a  Binomial  Surd  by  Inspection. 
By  actual  multiplication  we  may  find, 

(1/2"+  v/5)'  =  2  +  2T/10  +  5-7-r2l/m 

In  the  square,  7  +  2l/10,  7  is  the  sum  of  2  and  5,  10  is  the 
product  of  2  and  5.  Hence,  in  extracting  the  square  root 
of  7  +  21/10,  we  are  merely  required  to  find  two  numbers 
such  that  their  sum  is  7,  and  their  product  10 ;  extract  the 
square  root  of  each,  take  the  sum  (or  difference)  of  the  roots. 
This  may  readily  be  done  in  all  cases  where  the  numbers 
involved  are  small.     In  general. 

Transform  the  surd  term  so  that  its  coefficient  shall  be  2; 

Find  two  numbers  such  that  their  sum  shall  equal  the  ra- 
tional term,  and  their  product  equal  the  quantity  under  the  rad- 
ical ; 

Extract  the  square  root  of  each  of  these,  and  connect  the  results 
by  the  proper  sign. 


244  ALGEBRA. 

Ex.  Find  the  square  root  of  18  +  81/51 

18 +  81/5"=  18 +  21/80. 

The  two  numbers  whose  sum  is  18  and  product  is  80  are 
8  and  10. 


.1/18  + 81/5  =  i/S'+l/lD 
=  21/2"+ i/IO. 


EXERCISE 

:  87. 

Find  the  square  root  of— 

1.  17-121/2: 

8.  77-241/10. 

2.  23  +  4V15. 

9.  87-361/5: 

3.  35-121/6: 

10.  14  +  31/3": 

4.  9-61/2: 

11.  8-J56-1/2: 

6.  42  +  281/2: 

12.  5i  +  3l/H: 

6.  73-121/^. 

10a'  +  9  +  6( 

13.  4i-|l/3: 

7.  26  +  41/3D. 

14.  2m  +  2l/m''  — 7i", 

15. 

xVa^  +  l. 

Find  the  fourth  root  of— 

16.  28-161/3: 

19.  193-1321/2: 

17.  49  +  201/6: 

20.  H--I81/2: 

18.  97-561/3: 

21.  y  +  yi/6: 

Find  by  inspection  the  square  root  of — 

22.  3  +  21/2:  25.  23-6l/m 

23.  9-2l/li.  26.  18 -121/2: 

24.  21  +  12VW.  27.  7  +  41/3: 

28.  Prove  that  Va  ±:  VI  cannot  equal  Vc, 

29.  Prove  that  Va  cannot  equal  b  +  Vc. 


BADICALS.  246 

VI.  Solution  op  Equations  containing  Radicals. 

234.  Simple  Equations  containing  Radicals. 

Ex.  1.   Solve  VxT7  -  1  =  x. 

Transpose  terms  so  that  the  radical  shall  be  alone  on  one 
side  of  the  equation. 

V¥T7  =  x+l. 
Squaring,  a;'  + 7  =  «' +  2x  +  l 

.-.  2x  =  Q 

a;  =  3,  Root 

Ex.  2.   Solve  V^T~B"+  Vx  =  5. 

Transpose  terms  so  that  one  radical  shall  be  alone  on  one 

side  of  the  equation. 


l/«TB"=  d-Vx. 
Squaring,  a;  -}-  3  =  25  — 10  Vx  +  x 

.-.101/^=22 
51/^=11. 
Squaring,  25a;  =  121 

In  general, 

Transpose  the  terms  of  the  given  equation  so  that  a  single  radical 

shall  form  one  member  of  the  equations- 
Raise  both  members  of  the  equation  to  the  power  indicated  by  the 

index  of  this  radical; 

Repeat  the  process  if  necessary. 

235.  Fractional   Equations  containing  Radicals.     If  a 

radical  occur  in  the  denominator  of  a  fraction,  it  is  necessary 
to  clear  the  equation  of  fractions,  being  careful  to  multiply 
correctly  the  radical  expressions  involved. 


246  ALGEBRA. 

2 

Ex.1.    Vx-V^^-^=    y — -g- 
Vx  —  Q 

Multiply  by  Vx  -  8,  Vx^  -  8a:  -  a:  +  8  =  2 


Va;'^  -  8a;  -  a;  -  6 
a;2  -  8a;  =  a:^  -  12a;  +  35 
4a;  =  36 
a;  =  9. 


V^4-3        '6Vx  —  5 
Ex.  2. 


l/x-2      31/^-13 
Clearing  of  fractions,  3a;  -  4Vx  -  39  =  3a;  -  lll^i  +  10 

7l/i-49 
Vx  =  7 
a;  =  49. 

EXERCISE   88. 

Solve  the  following  equations : 

1.  1/^+1  =  3.  12.  2T/3F=^  =  3l/i'Tl. 

2.  1/^+1  =  3.  13.  SVx-l=  Vx  +  1, 

3.  1/35-2  =  1.  14.  VTTT6  =  8-1/^, 

4.  5-l/2i  =  3.  15.  1/^  =  3-1/^^=^. 

5.  1  =  1/35^=3.  16.  l/5^=n["5  =  15  -  l/i. 

6.  V2x  +  d  =  2.  17.  I  +  1/5  =  l/-i^^. 

7.  1  =  1^5-1.  18.  f  -  1/25=  1/25+^. 


8.  1/2x^1  +1=4.  19.  1/5  +  1/5"+^  =  8. 

9.  3  =  2-l>'35Tl.  20.  1/45+^  =  21/5^1  +  1. 
10.  a; -  1  =  l/ST^.                21.  21/5-  1/45^=22  =  l/2. 


11.  l/a;  +  l=l/13.  22.  l/9a;  +  35  =  7 1/5 - 3 V « 

23.  ^13  +  ^V  +  1/3+1/5  =  4. 


RADICALS.  247 

25.  l/25^^"29  -  Vi^rnj  =  3 1/i. 

26.  V^+  V1mTx  =  2\/FTx. 

28.  ^^^+"^4-  V^  = 


VT=R; 


29.  l/'9T2x  =  -— J:=r  +  |/2i. 
Vd  +  2x 

SO.  2Vx-VW=^=        ^ 


T/i^^'H 


31.  3l/2STT-3i/2^^^  = ^ 

32.  ^-^'      l^^+l 


l/a;  +  3       l/a;-2 
Vx  +  2       Vx  +  7 


35. 


V^-1        l/a;  +  l 

6v^-7       ,      7l/i-26 
34.  — 5=  — 

V^-1  7V^-21 

l/^~a  +  Vx 

— == =  a. 

Vx  +  a—Vx 

VWn  +  Vox  1 

36.  — ==^ — —  =  1  +  t;- 

x  —  a  Vx—  Va      ^    _ 

37.  —= = . +  2  V^ 

Vx+Va  ^ 

VWT2-Wx      , 

38.  ^  -=4. 

VWT2  +  3V^ 


248  ALGEBRA. 


89.  ^2^  +  ^1^  +  1/^+1/^=1/5. 

40.  Vx  +  Va  —  V  aS  +  cc"  =  l/a. 
a  +  3         a  -  3  2a 


41. 


1^  +  2      1^  —  2      a;  — 4 


42.  J^- J? -1  =  1. 


1,1  /I 


43.  -  +  -  =  \/-  +  v'^  + 


/XTT 


X      b       ^25 


+  V;i^ 


44.  Va:'^  +  4a;  +  12  +  Va;-^- 12a; -W=8. 

45.  l/4xM="6ar+T-  l/4a;-^  +  2a;  +  3  =  1. 

46.  l/9a;'^  +  a;  +  5-  V 9a;^  +  Va;  +  6  -  1=0. 

236.  Summary  of  Principles  relating  to  Radicals.     Let 

the  student  form  a  summary  of  the  principles  relating  to 
radicals,  similar  to  tnat  given  in  Art.  206  for  exponents. 
Thus, 

Transformations  of  Radicals. 

I.  Simplification  of  Quantity  under  the  Radical  Sign> 

1.  General  Case.    Ex.  l/8a^=  2a  i/2a6 

2.  Fraction  under  Radical.    Ex.  l/f  =  -J  l/B. 

Etc.,  Etc. 

EXERCISE  89. 

REVIEW. 
1.  Write  in  the  simplest  form — 


1         1         2 

1                 2 

17                  3                   12 

v%-i '  2-1/5' 

1/^-2'    V5+V2'    1/7-1'    3V^-21/5' 

REVIEW.  249 

Collect— 

2.  2Vn  -  1/75  +  1/18  -  1/2?  +  1/5; 

3.  §1/45  -  30V^  +  f  1/20  -  61/-^  +  1/500. 

4.  4V'n7  -  31/75  -  6l/|^  +  18VJj  -  24Vf. 

'  36\l6a*        a''\56'''       a26\49a;       2a  \  20a;' 

Multiply — 

6.  2  +  1/St  -  l/g"  by"  2  +  1/3"  +  VE. 

7.  1/a  +  l/oTl;  -  l/i  by  l/a  -  l/aT~a;  -  l/£ 

8.  ^l/f  by  3l^^7;   and  2l^|f  by  il^W- 

Divide — 

9.  61/I2  +  31/8  -  6V30  +  4VI5  by  2V^. 

10.  f  1/70  -  f  1^28  +  3VI05  by  f  1/42. 

11.  a:^  _  a;  -f  1  by  a;  +  1/a;-  1. 

Rationalize  the  denominator  of— 
12   2  + V^  .  14   6l/g  -  41/g. 

*  3  -  1/^  '  5l/r+  31/^ 

j3   21/3  -  31/^.  jg   21/15  -f  8  ^  81^3  -  6Vo_ 

3  -  21/5  *  5  +  Vl5       51/3  -  Zi/S 

Find  the  numerical  value  of— 

16.  -^  •  18  21/5"- 31/g. 

V^  '  31/5  -  41/2 

17    V^-T^  .  19.  31/6  -  21/?. 

'  21^7  +  1/2  *     41/2  +  5 

Which  is  the  greater — 

20.  31/3  or  2VT?  22.  2l/6^  or  31^5? 

21.  1/8  or  1^23?  23.  V^  or  l^'U? 


350  ALOEBBA. 

Find  the  square  root  of — 

24.  33  +  201/2.  26.  80  -  32V1S". 

25.  35  -  12VE,  27.  107  +  121/77. 


Simplify 

28.^^1 

Vx  -1      x  +  Vx 


a^+l'-i/^-i       T  +  T/^i 


1  + 


1 


29. ^f^-  30.{V|-2-Vf^}*- 


1/1 -x» 


Solve— 


31.  xJ^^  +  2/V-^^^  -  (3y'  -  *')  V*?^ 


32.  2  +  l/x  +  3  =  1/a;  -  2  +  3. 

33.  i  -  l/^^r2  =  1/^. 
1/5 


3^» 


34.  Vx  +  Vx^^^ 


v^r^ 


gg    1/^  +  4         1/a;  +  2 


21/5  -  1       21/5  -  3 

36.  21/35^F1  =  31^3^  +  5l/3F=TL 

l/3a;  +  4 


CHAPTER    XVIII. 
IMAGINARY  QUANTITIES. 

237.  An  Imaginary  Quantity  is  an  indicated  even  root  of 
a  negative  quantity. 

Exs.  1/^=^     i/^=T,     V^H, 

The  term  "  imaginary  "  is  used  because  so  long  as  we  confine 
ourselves  to  plus  quantity,  and  to  its  direct  opposite,  minus 
quantity,  there  is  no  number  which  multiplied  by  itself  will 
give  a  negative  number,  as  —  4,  for  instance.  All  the  quan- 
tity considered  hitherto  that  is  plus  or  minus  quantity, 
whether  it  be  rational  or  irrational,  is  called  real  quantity. 

If  we  extend  the  realm  of  quantity  considered,  outside  of 
plus  quantity  and  its  direct  opposite,  minus  quantity,  imag- 
inary numbers  are  as  real  as  any  others,  as  will  be  shown  in 
the  next  article. 

A  number  part  real  and  part  imaginary  is  called  a  Com- 
plex Number. 

Exs.  3  +  2l/^n[,    a  +  61/^=^1. 

If  an  imaginary  number  exist  by  itself,  it  is  called  a  Pure 
Imaginary.    Thus,  3  V^^l  is  a  pure  imaginary. 

238.  Meaning  of  l/^=l. 

Let  us  consider  the  simplest  imaginary,  V—l^  and,  by  a 
geometrical  illustration,  try  to  discover  how  it  has  a  mean- 
ing if  we  extend  the  realm  of  quantity  outside  of  plus  quan- 
tity and  its  direct  opposite,  minus  quantity.  If  OA  =  +  1, 
and  OA'  be  of  the  same  length,  but  lying  in  the  opposite 
direction  from  0,  OA'  —  —  1. 

Hence,  we  regard  the  operation  of  converting  a  plus  quantity  into  neg- 

251 


252 


ALGEBRA. 


^v^ 

0 

.  f 

1 

-1 

+  1 

-V--1  -j 

ative  quantity  as  equivalent  to  a  rotation  through  an  angle  of  180°.  If  we 
divide  this  rotation  into  two  equal  rotations,  each  of  these  will  be  a  rota- 
tion through  90°.     But  in  seeking 

a  square  root  of    —1    we    seek   a  B 

factor,  l/—  1,  which  multiplied  by 
itself  will  give  —1.  The  result 
of  a  rotation  of  -f  1  through  90°, 
rotated  again  through  90°,  gives 
-1. 

Hence,  V—  1  must  be  equivalent 
(geometrically)  to  the  result  of  ro- 
tating the  plus  unit  of  quantity 
through  90®.  Hence,  V-  1  on 
our  figure  will  be  represented  by 
OB. 

Hence,  it  is  easy  to  see,  also,  that  B^ 

V^^l  X  V^^  =  -1.     We   thus 

perceive  that  the  introduction  of  imaginary  quantity  enlarges  the  field  of 
quantity  considered  in  algebra  from  mere  quantity  in  a  line  to  quantity  in 
a  plane.  This  gives  a  vast  extension  to  power  of  algebraic  processes  and 
introduces  many  economies  in  them,  as  will  be  found  by  the  student  who 
pursues  the  study  of  mathematics  extensively. 

In  taking  up  the  subject  for  the  first  time  we  consider  only  a  few  of  the 
first  properties  of  imaginaries,  so  called. 

239.  Fundamental  Principle.  We  regard  all  the  ordinary 
laws  of  algebra  as  applying  to  imaginaries,  but,  owing  to  the 
nature  of  a  square  root,  one  modification  of  these  laws  as 
ordinarily  stated  must  be  made. 

This  fundamental  new  principle  is  that  V  —  1  X  V^^^l 
=  -1. 

Besides  the  above  geometrical  illustration  (Art.  238),  it  is 
important  to  state  formally  the  algebraic  reason  for  this 
principle. 

The  square  of  V  —  1  must  be  such  a  number  that  when  its 
square  root  is  extracted  we  shall  have  the  original  quantity, 

If  we  use  the  law  of  signs  in  the  most  general  form, 
( V^=~l/  =  V^l  X  V^^l  =  l/r=  zt  1. 


IMAGINARY  QUANTITIES.  253 

Now,  if  we  extract  the  square  root  of  +1,  we  shall  not 
have  V^^^l.  But  if  we  extract  the  square  root  of  —  1,  we 
shall  have  V—  1. 

Hence,  we  must  limit  the  product  V  —  1  X  V^^^  to  —  1. 

Likewise  V^^X  V~^^  =  VaV^^l  X  VhV^^l 

or,  in  general, 

The  product  of  two  minus  signs  under  a  radical  sign  of  the 
second  degree  is  a  minus  sign  outside  of  the  radical. 

240.  Reductions  to  the  Typical  Form  A  -{-  BV^~^1.  It 
may  be  shown  that  any  imaginary  expression  of  any  degree 
of  complexity  can  be  reduced  to  the  form  a  +  bV  —  1.  We 
will  limit  ourselves  at  this  time  to  showing  that  any  com- 
bination of  the  sum,  difference,  product,  or  quotient  of  im- 
aginary expressions  of  the  second  degree  reduces  to  this 
form. 

First.    Sum  or  difference  of  two  complex  numbers. 

a  +  6l/^T  ±  (c  +  dV^^  =  (a  i  e)  +  (6  ±  d)V^^ 
=  ^  +  BV=~1 

Second.    Product  of  two  complex  numbers. 

(a  +  bV^^  (c  +  dl/^T)  =  ae-bd  +  {be  +  ad)V^=T. 

^  A'  +  B'v^=n: 

Third.    Quotient  of  two  complex  numbers. 

c  +  dV^  _  c  +  dl^^^T  ^  a-  h\^^^ 

a  +  6V^^=T      a  +  6i/^=T:      a  -  bV^^ 

go  +  bd  +  (ad  -  6c)V-  1 

a^  +  b^ 
ac  -f  bd    ,   ad  -  be  ^y—^ 

=  A''  +  b'^i/^=t: 

Hence,  any  combination  of  the  sum,  difference,  product,  and  quotient 


254  ALGEBRA. 

of  complex  numbers  may  be  reduced  by  successive  steps  to  the  typical 
form. 

This  serves  to  illustrate  the  fundamental  value  of  the  imaginary  unit, 


241.  Powers  of  V^  —  1. 

{V^=^y  =  - 1,  .  • .  ^2  =  - 1. 

{y^^f  =  {V^^^YV^^  =  -  V~l,     ..e  =  -i.  (See OJl^of figure 
{V^Ty  =  {V~-^iyV~i  =  +  l,  .  • .  i*  =  l.  in  Art.  238.) 

The  symbol  i  is  used  for  1/—  1. 

Thus  the  first  four  powers  of  V-  1  are  T/-  1,  —  1,  —  V^HT,  +  1 ; 
and  for  the  higher  powers,  as  the  fifth,  sixth,  etc.,  these  four  results  recur 
regularly.    The  same  fact  is  plain  from  the  figure  in  Art.  238. 

242.  Equational  Properties  of  Imaginaries.  I.  If  an  im- 
aginary expression,  ac  +  yV  —\,  equals  zero,  then  x  =  0,y  =  0. 

Proof,  x  +  y  V  —  1  =  0 

,-.  x^  +  y'^O, 

which,  since  «  and  y  are  both  real,  can  be  true  only  when 
x  =  0,y  =  Q. 

II.  If  two  imaginary  expressions  he  equal,  the  real  part  of  one 
equals  the  real  part  of  the  other,  and  the  imaginary  part  of  the  one 
equals  the  imaginary  part  of  the  other. 

Proof     Let  x  +  y  V^^l  =a  +  h  V^^l. 

.'.  x-a  +  (y-b)\/^=n.=0. 
.  * .  by  I.  of  this  Art.,  x  —  a=0,        .' .  x  =  a, 

y-b=0,        .' .  y  =  h. 

243.  Conjugate  Imaginaries.     If  two  complex  numbers 


IMAGINARY  QUANTITIES.  255 

differ  only  in  the  sign  of  their  imaginary  part,  they  are  called 
conjugate  imaginaries. 

Exs.  3  —  i/"=l  and  3  -f-  i/^T; 
a-\-b  i/  —  c  and  a—b -y/  —  c. 

244.  Operations  with  Imaginaries.  It  follows  from  Art. 
239  that,  in  performing  operations  with  imaginaries,  we 

Use  all  the  ordinary  laws  of  algebra^  with  the  exception  of  a 
limitation  in  use  of  signs,  which  may  be  mechanically  stated 
as  follows :  The  product  of  two  minus  signs  under  the  radical 
sign  of  the  second  degree  gives  a  minus  sign  outside  the  radical 
sign.  But  in  dividing  first  indicate  the  division  and  after- 
wards rationalize  the  denominator. 


Ex.1.    Addi/— 9,  — 3  +  2i/— 1,  7  — 2i/=^16; 

v^^=         3  i/^^nr 

—  3+2  i/^^~l=  —  3  +  2  i/"^^! 
7_  2  1/^^16=       7_8i/"=ri 


4  —  3  i/—  1,  Sum. 


Ex.2.  Multiply2i/— 3  +  3i/— 6by3i/— 3— 5i/— 27 

2i/^=^+3i/=6 

3t/=^— 5i/^^ 

6(-3)— 91/18 

+  10  i/6  +15t/T2 


18—27  i/2  +  10  i/  6  +  30  >/  3,  Product, 


Ex.3.    Divide  —  2  ]/  6  by  i/— 2. 
—  2  i/'6      — 2  v/'6       y^=^ 


l/  — 2  i/- 2         i/— 2 

— 2i/'=12  , ^ 

=  — ^^72 —  =  2i/— 3,  Quotient, 


Ex.  4.   Extract  the  square  root  of  1  +  4 1/ — 3. 
1+4 1/— "3  =  1  +  2 1/^=12: 
The  two  numbers  which  multiplied  together  give  — 12,  and  added 
together  give  1,  are  4  and  — 3. 


.  l/l  +  4i/— 3  =  1/4  +  1/—  3  =  2  + 1/—  3,  ^esttW. 


256  ALGEBRA. 

EXERCISE   90. 

Reduce  to  the  form  aV^^l. 

1.  I/- 9:  3.  v-im.  5. 2V~=i;. 

2.  1/^-25".  4.  -  V^=^.  6.  -ZV^=^. 

7.  -Ji/^=^SB:  8.  11/^=^324: 

Collect — 

9.  7  V'^^  +  3 1/^^=^^- 10  i/^^a 

10.  2V^T-ZV- 121  +  51/^^4. 

11.  l/'='400  +  2l/~  900-51/-  144. 

12.  1/^:^:2  -  T/"~27'+  21/^^+  1/^=^75: 

13.  5  V~^^-  3  V^=T+  4 1/^^50"-  1/-200. 

14.  2l/"=^^-3al/'^^T-r  -V  -  16a*  - 1 1/ -  36al 

15.  a  +  6l/^=T-6-al/"=T- l/'=^  +  2l/^=^^-a. 

16.  (a-26)l/'^=^~(2a  +  6)l/^=T: 

Multiply — 

17.  1/^=1:  by  1/^^^  22.  y^=^by  -31^^=^ 

18.  2v"=^by  31/^=^  23.  2 l/"=T4  by  -  2 y^2r 

19.  V^=l>  by  -2V^=^b.  24.  -bV^=2  by  - 21^^=^: 

20.  -  1/^=^  by  —  l/"=T  25.  —  V^^  by  Vy^=^. 

21.  -  i/=T:2  by  -  T/=T8.  26.  - aVY=^a  by  -  l/(a  -  1)' 

27.  V^=n.  +  V~=^hy  l/^=T- 2 1/^^2: 

28.  3l/=^- 2 1/'^=^2" by  21/^=^+31/"=^ 

29.  2l/2-2l^=^by  3l/2  +  3l/'^^2: 

30.  ZV^^-  2l/^=^by  41/"=^+  SV^^^ 


IMAGINARY  QUANTITIES.  257 

81.  l-3v^^^Sby  1+51/^^^^. 

32.  V=^-V^=^2+  V^^  by  V^^+  V^=^+  1/^=^ 

38.  x  -  2  +  1/^=^  by  a;  -  2  -   V^^^. 

34.  al/  —  a  +  61/^=^5  by  al/^=^— 6l/^=T. 

35.  a;  -  1  —  V^=n.  by  a;  -  1  +  V~^l. 

1-V^=^  ,  1  +  V^=^ 

36.  X by  a; 

2  -^  2 

Divide — 

37.  -vlSby  V^^.  39.  -  6 l/^T5  by  2 1/"=^. 

38.  -V^^^T2  by  -1/"=^.         40.  8V"=^  by  -2v/a. 

41.  2V=n^-AV'=lF+  lOl/BOby  -2i/-'3. 

42.  a  1/"=^  -  2a  l/^^-  a VBc?  by  -  a  l/^=^. 

Express  with  rational  denominators — 

1                                         SV^=~5-V~^^ 
43. 47. 

3-l/^"2  2v^^^-3l/-2 

2  -  1/=^  l/T="l  +  VT=^ 

44. 48. 


2  +  V^=B  2l/r=^- 1/^"^=! 

45.  -— •  49.  -^ ;=/-• 

23/2+1/^=^  2-3l/^=nL 

a  +  ftl^=n[  3v/2  +  2l/"=^-l/^^ID 

46. .  50. " 

a-bV^=A  31/2-21/^^  +  1/^=10 

Find  the  square  root  of— 

51.  S-QV^e:  64.  121/ID-38. 

52.  1-21/^=^6:  65.  -29 -241/=^ 

53.  12 1/==^- 6.  56.  7  +  40l/^=T. 

17 


258  ALGEBRA. 

57.  Find  the  value  of,  (v"=^)';  (-1/^=1)*;  (V^^^l)*; 
(-l/-iy;  (l/"-=l)-^  (-V'--l)-^  (1/"=^)* 

In  expanding  binomials  containing  imaginaries,  the  labor  is  perhaps  less- 
ened by  the  substitution  of  a  letter,  as  i,  for  the  imaginary,  V^^l,  and 
after  simplifying  this  derived  expression,  in  i,  return  to  the  imaginary, 
V'-l. 

Fot  example,  let  it  be  required  to  simplify  the  expression,  3(1/—  1  +  2)^ 
-  (2l/~T  -  1)2.     Substitute  i  for  V~=i:. 

3(i  +  2)2  -  (2i  -  1)2  =  3i2  +  l2i  +  12-4*2  +  4^-1 

=  -  i2  +  16i  +  11 

=  -  {v^^y  + 161/^^  + 11 

=  12  +  161/^=^",  Eesult 

Simplify — 

68.  (1/^1-1)' -(1/^=^-1^  +  2(1/^=1-1). 

69.  (1/=1  -  2)  (3  V^l  +  1)  -  (1/^=1  -  By  -  (l/==^\ 

60.  (  V^l  -  1)*  +  3(  1/=~1  -  1)'  +  4(  l/'^I  -  1)1 

61.  Prove  that  the  sum  and  the  product  of  any  pair  of  con- 
jugate imaginaries  are  real.     (See  Art.  243.) 

62.  li  x=  ^~  ^^""^  ,  find  the  value  of,  3x'  -  Ox  +  7.     0£ 

2 

s?-6x'  +  2x-l. 

63.  If  a;  =  ^"^^^^^  '  find  the  value  of,  lOa;*  -•  8x  +  3. 


CHAPTER    XIX. 

QUADRATIC  EQUATIONS  OF  ONE  UNKNOWN 
QUANTITY. 

245.  General  Problem.  The  relation  of  the  square  of 
some  unknown  number  (as  well  as  of  its  first  power)  to 
known  numbers  may  be  given  in  the  form  of  a  more  or  less 
complex  equation.  It  then  is  often  required  to  reduce  this 
complex  relation  to  some  simple  relation  from  which  the 
value  of  the  unknown  number  may  be  at  once  recog- 
nized. 

246.  A  Quadratic  Equation  of  one  unknown  quantity  is 
an  equation  containing  Ihe  second  power  of  the  unknown 
quantity,  but  no  higher  power. 

A  Pure  Quadratic  Equation  is  one  in  which  the  second 
power  of  the  unknown  quantity  occurs,  but  not  the  first 
power. 

Ex.  5x^-12=0. 

An  Affected  Quadratic  Equation  is  one  in  which  both 
the  first  and  second  powers  of  the  unknown  quantity  occur. 

Ex.  3a;^- 7a; +  12  =  0. 


PURE  QUADRATIC  EQUATIONS. 

247.  Solution  of  Pure  Quadratics.  Since  only  the  sec- 
ond power,  x',  of  the  unknown  quantity  occurs  in  a  pure 
quadratic  equation, 

Reduce  the  given  equation  to  the  form  x^  =  c; 
Extract  the  square  root  of  both  members. 

25» 


260  ALGEBRA, 

Ex.1.   Solve  — - —  =  —- — 

Clearing  of  fractions,  4x'  —  48  =  Sx'^  — 12 
Hence,  x"  =  36 

Extracting  the  square  root  of  each  member, 
a;  =  +  6,     or  —  6. 

That  is,  since  the  square  of  +  6  is  36,  and  also  the  square  of 
—  6  is  36,  X  has  two  values,  either  of  which  satisfies  the  orig- 
inal equation.  These  two  values  of  x  are  best  written  to- 
gether.    Thus, 

a;  —  ±  6,  Roots. 

Ex.2.  Solve 


x'-b 

~  x' 

—  a 

ax'- 

-a'  = 

bx'-b^ 

ax'- 

6x^- 

■a'-b' 

X  — 

^a  +  b 

--±Va  +  b, 

EXERCISE  91. 

Solve — 

1.  5x^-80.  g   J L=  5 

2.  3x^-5=a;'  +  3.  *   4x^       Sx'  ~^' 


9. 


3.  ix'-l=i-Sx\ 

4.  l-|x'=-x^-4f.  2x-l        2a; +  1 

x"  ,      ^  10.  ax'' +  a' =  5a' —  3ax'. 

5.  — -4  +  x'  =  0. 

8       ^  11.  ax'  +  c^b. 

^   ^Z:5_|^-x\  ^2.  ^  +  2a    ^  x-2a^,^,^ 


5  X  —  2a       X  +  2a 

U  6  x-c         3x 


^_3^^^+5^3_  i3._2^+5x±2c^^^^^ 


QUADRATIC  EQUATIONS.  261 

14.  (ax  4-  by  +  (ax  -  bf  =  10b\ 

15.  (x  +  a)(x-b)  +  (x-a)  (x  +  6)  -2(a'  +  6»  +  ab). 

16.  3(2a:-5)(a;  +  l)-2(3a:4-2)(2a:-3)-a;-9. 

AFFECTED  QUADRATIC  EQUATIONS. 

248.  Completing  the  Square.  An  affected  quadratic  equa- 
tion may  in  every  instance  be  reduced  to  the  form 

x^  +  px  —  q. 

An  equation  in  this  form  may  then  be  solved  by  a  process 
called  completing  the  square.  This  process  consists  in  adding 
such  a  number  to  both  members  of  the  equation  as  will  make 
the  left-hand  member  a  perfect  square.  The  use  of  familiar 
elementary  processes  then  gives  the  values  of  x. 

Thus,  to  solve  x''  +  Qx  =  16, 

take  half  the  coefficient  of  x  (that  is,  3),  square  it,  and  add 
the  result  (that  is,  9)  to  both  members  of  the  original  equa- 
tion.   We  obtain 

a:'  +  6a;  +  9  -  25. 
Extract  the  square  root  of  both  members, 

a; +  3  =  ±5, 
Hence,  a;  =  —  3  ±  5, 

That  is,  a;  =  -  3  +  5  =  2, 


Also,  x=  — 3  —  5=  —  8, 


a 


Hence  we  have  the  general  rule : 

By  clearing  the  given  equation  of  fractions  and  parentheses, 
transposing  terms,  and  dividing  by  the  coefficient  of  x^,  reduce 
the  given  equation  to  the  form  x^  +  px  =  q; 

Add  the  square  of  half  the  coefficient  of  x  to  each  member  of 
the  equation; 

Extract  the  square  root  of  each  member; 

Solve  the  resulting  simple  equations. 


•262  ALGEBRA,^ 

Ex.  1.   Solve  Gx'  -  14a:  =  12. 
Dividing  by  6,  x^  —  ^x  =  2 

Completing  the  square,  a;'  -  |x  +  (|)'  =  2  +  ff  =  W* 
Extracting  square  root,  a;  —  J  =  dz  ^ 

ic  =  3,    or  —  I,  i?oo^. 
Ex.2.   Solve  3x^  =  2(1  + 2a:). 
Clearing,  3a:'  =  2  +  4x 

Transposing,  3x'  — 4x  =  2 

Hence,  a:^  ~  |a^  =  t 

x  =  i±VW  =  ^^^^  Roots, 

o 

EXERCISE   92. 

Solve — 

1.  x^  +  14a:  =  32.  14.  35  -  2a:^  -  3ar. 

2.  x^  +  lOx  =  24.  15.  3a:  +  77  =  2x\ 

3.  a:'-8a:-20  =  0.  16.  6a:' -  a:  -  35  =  0. 

4.  x'-5a:  =  6.  17.  3x'  +  ia:  =  l|. 
6.  x^  +  11a:  +  24  =  0.  18.  3a;'  =  ix  +  2|. 
6.3.^4-4x^7.  I9.i  =  ^  +  x'. 


7.  5x'-6x  =  8. 


6 


8.  2x^-5x--7.  20.  ^^  +  -^-  =  2J. 


9.  3x'^  +  7x-26. 


2  x-1 


10.  4x^-h8x-5  =  0.  21.  ^-1  +  — ^=0. 

ox  a:  ~r  2 

11.  6^-5x-6  =  0.  ^„    3x  +  5_,      2.-5 

12.  2.  +  3|a:'  =  4.  ^-    ^  +  4   ~  x-2  ' 

13.  a:'  +  5  =  ^.  23.  ^^^^  - -^±i  =  -  i 

3  x  +  S       2.-3  ' 


QUADRATIC  EQUATIONS. 


31.  i(^  +  l)-|(2«-l)  = 

„„    2x-l       Zx-4  _ ,      4a; -14 

x  +  1         x-1  l-x" 

x-B        hx-1        2a;  +  5 
o4.    _  _   + 


3x-2      4-9a;='       2  +  3a; 

„,     2a;-l      l-3x       x-1      , 
35. =  4. 

a;  +  l         a;  +  2        x-1 

36.  a;'  +  2a;  =  l.  40.  lla;»- 12a;  = -3. 

37.  3x'-5x=-l.  41.  2a:'  +  5a;=-4. 

38.  9ar^-18a;  +  4-0.  42.  3a;'-7a;=-5. 

39.  5x'  +  3a;  =  l.  43.  9x'-6a;  +  5=0. 

44.  3a;(a;  +  l)-(a;-2)(a;  +  3)  =  2H-(l-a;)'. 

45.  (x  +  l)(x-5)-2a:(a;-l)  =  l-(l-2a;)*. 

46.  ^^-i  +  ^^:^^2. 

x'  +  x+l       a;'  — a;+l 


N.  B.    The  verification  of  equations  having  irrational  roots  is  a  profitable 
ercise. 

249.  Literal  Quadratic  Equations  are  solved  by  the  same 


exercise 


264  ALGEBRA. 

methods  employed  in  solving  quadratic  equations  with  nu- 
merical coefficients. 

Ex.  1.   Solve  ^-^=%  +  a). 
2        6       6^ 

Clearing,     Sx'^  -  ax  =  ax  ■¥  o? 

Hence,      ^x"^  —  lax  =  a^ 

3  3 


^-().(ff  =  if 


^    Boots, 


3 

Ex.  2.  Solve  (a  -  6) V  -  (a'  -  6^^^  =  -  a6 


,2  _  «_±J'a;  = ^^ 


(a  -  6)^ 
a  +  b  ,      a-h 


2{a-b)  2{a-b) 

X  =  —^  ,    -^—  ,  Roots. 
a  —  b      a  —  b 

EXERCISE   93. 

Solve— 

1.  a:'  +  4ax  =  12a^  10.  2aV  +  ahx  =  156\ 

2.  a;'^  +  46a;  =  216^  11.  x'  -  (a  +  l)a;  =  -  a. 

12.  x'  H = 

4.  x'  +  bahx  =  Q>a'b\  «         4^2 

5.  6a:'  -hx  =  12b\  13.  x'  +  (2  -  3a)a;  =  Ga. 

6.  3a;' +  4ccZa;  =  ISc'dl  14.  3aV  +  a(36  -  5)a;  =  56. 

7.  2aV  +  aa:  =  3.  15.  abx'  +  (a'  +  6')a;  +  a6  =  0. 

8.  7cV  ~  lOaca;  +  3a' =  0.  16.  ax' -  (a' —  l)a;  =  a. 

a       a'  x'  a' 


QUADRATIC  EQUATIONS.  265 

18.  X —  =  a.  19.  4(x'  -  1)  =  6(4x  -  6). 

X  —  a 

20.  Ca-{-b)x'-(a-b)x ^=0. 

a  +  b 


21.  abx"  =  -^  fxCa  +  6)  -  — 1  • . 
a6  L  abj 


x  c  a  +  b 

23.  a(a:'  -  b')  +  b(x'  -b^-tc)  +  cx  =  0. 

24.  (a  +  c>'  — (2a  +  c)x  +  a  =  0. 

FACTORIAL  METHOD  OP  SOLVING  EQUATIONS. 

250.  Factorial  Method  for  Solving"  Quadratic  Equa- 
tions. If  any  factor  of  a  product  equals  zero,  the  entire 
product  equals  zero.  Hence,  if  a  quadratic  equation  be 
reduced  to  the  form  ax^  +  bx-{-  c  =  0,  and  the  left-hand  mem- 
ber be  factored,  and  each  factor  be  made  equal  to  zero,  the 
values  of  x  thus  obtained  will  satisfy  the  equation,  and 
therefore  be  its  roots. 

Ex.  Solve  a;'  +  5a;  —  24  =  0. 
Factoring,  '    (ck  +  8)  (x -3)  =0. 

Hence,  letting  a;  +  8  =  0,        and  a;  —  3  =  0, 

a;  =  —  8,        a;  =  +  3,  Roots. 

251.  Factorial  Solution  of  Equations  of  Higher  De- 
grees. Since  the  principle  of  Art.  250  applies  to  the  prod- 
uct of  any  number  of  factors,  equations  of  degree  higher 
than  the  second  may  often  be  readily  solved  by  this  method. 

Ex.1.   Solvea:(a;-l)(a:  +  3)(a;-5)=0. 
The  roots  are  a;  =  0,  1,  —  3,  5. 


266  ALGEBRA, 


Ex.2. 

Solve  x'  +  1 

-0. 

Factoring 

r,    {x  +  l){x'- 

-  X  + 

1)  = 

-0 

x+  1  = 

=  0, 

gives 

X 

==- 

1,  Root, 

Also, 

x' 

-  X  - 

f  1  = 

=  0 

Whence, 

x"- 

-  X  = 

=  — 

1 

X  - 

-I 

i 

\^^~ 

-  3,  Roots. 

Ex.  3.  Solve  x\  +x'-  4(x'  - 1)  =  0. 

Factoring,  x\x  +  1)  -  4(a;'  -  1)  =  0 

(a:  +  1)  {x"  -  4a:  +  4)  =  0 
(a;  +  1)  (a;  -  2)  (a:  -  2)  =  0 
a;.-  -  1,    2,     2,  -Boote. 

EXERCISE   94. 

Solve  by  factoring — 

1.  a;'  +  8x  +  7=0.  11.  a:*- 5x' +  4  =  0. 

2.  a;'-5a;  =  84.  12.  a;^-a;*-a;  + 1  =  0. 

3.  Bx''  -  X  -  15.  13.  (2x  -  1)  (Bx'^  -  x  -  2)  =  0 

4.  Ba:^  +  7a;-90.  14.  3(x' -  1) - 2(a;  +  1)  =  0 
6.  12x^  -  5a:  =  3.  15.  5(a:^  -  4)  =  3(3:  -  2). 

6.  3a:^  -  lOa:  +  3  =  0.  16.  7(a:*  - 1 B)  -  53a:(a;^  -  4)  -  0. 

7.  24a:^  =  2a;  +  15.  17.  3x(ar^  -  1)  +  2(a:  - 1)  -  0. 

8.  3aV  +  lOax  =  8.  18.  a;'  -  27  =  13a;  -  39. 

9.  a:*  =  IB.  19.  2a:''  +  2x'  =  a:  +  1. 

10.  a^  -  8.  20.  2a:'  +  Bx''  =  3a;'  +  8x  -  3. 

21.  Find  the  six  roots  of  x"  -  1  =  0. 

EQUATIONS  IN  THE  QUADRATIC  FORM. 

252.  Simple  Unknown  Quantity.  An  equation  contain- 
ing but  two  powers  of  the  unknown  quantity,  the  index  of 
one  power  being  twice  the  index  of  the  other  power,  is  aa 


QUADRATIC  EQUATIONS.  267 

equation  of  the  quadratic  form.     It  may  be  solved  by  the 
methods  already  given  for  affected  quadratic  equations. 

Ex.  1.   Solve  x*  -  5a;' =- -  4. 

Adding  (f )'  to  both  members  will  make  the  left-hand  member  a  perfect 
square,  giving 

X*  -  5a;2  +  (f  )2  =  f 
Hence,  ic''  —  f  =  =t  f 

x^  =  4,     or  1 
x  =  ^2,     ±  J,  Roots. 

This  equation  might  also  have  been  solved  by  the  factorial 
method. 


Ex.2. 

So\ye2v'x-'-Sv'x-'  =  2. 

Using  fractional  exponents, 

2x~^-Sx~^  =  2 

Whence 

x~^-ix~^  =  l 

a:"^-()  +  T\  =  M 

a;-^-f  =  ±f 

x-^  =  2,     -1 

Whence 

x^  =  h     -2 

x  =  l     -  8,  Roots. 

253.  Compound  Unknown  Quantity.  A  polynomial  may 
be  used  in  the  place  of  a  single  quantity  as  an  unknown 
quantity. 

Ex.1.   Solve(2a;-3)'-6(2a;-3)=7. 
Let  2a;  —  3  =  ^,  and  substitute. 

We  obtain  ^/^  -  6^  =  7 

Whence  2/  =  7,     -  1. 

Hence,  2a;  -  3  =  7,        also,     2a;  -  3  =  -  1 

.  • .  x  =  5,  Root.        a;  =  1,  Root. 


268  ALGEBRA, 

Ex.  2.  Solve  V^rri2  +  v'FTT^  =  6. 

1  1 

This  equation  may  be  written,  {x  +  12)^  +  (a:  +  12)^  =  6. 

Let  {x  +  12)^  =  ^  ;        then  {x  +  12)^  =  y^ 

Hence,  substituting,         2/^+^  =  6. 
Whence  2/  =  2,     or  —  3. 


.  • .  1/a:  +  12  =  2,  Also,  l/a;  +  12  =  -  3, 

a;  +  12  =  16,  a;  +  12  =  81, 

X  =  4,  Boot.  X  =  69,  BooU 

Ex.  3.   Solve  x\ - 7a;  +  Vx'-lx  +  l^^ 24. 

Add  18  to  both  sides, 

a:2  -  7a;  +  18  +  Vx'  -  7a;  +  18  =  42 
Let  Vx'^  -  7a;  +  18  =  ^  ;        then  y"^  +  y  =  42 

y  =  6,     or  -  7. 
Hence,  l^a:'' -  7a;  +  18  =  6,  Also,  1/a;"^  -  7a;  +  18  =  -  7, 

a;2  -  7a;  +  18  =  36,  a;^  -  7a;  +  18  =  49, 

a;  =  9,     -  2,  Boots.  x  =-•  1(7  =1=  VlTB),  iJoote. 


EXERCISE   95. 

10.  9a;"*  +  4  =  13a;~*. 

n.  Zv^-Wx  =  -2. 
12.51^^=81^^  +  4. 

13.  7l^^^-4l'/i^  =  3. 

14.  3  l/2i- 21^2^  =  1. 

15.  (a;-iy  +  4(a;-l)=21. 

16.  2(a;^-3y-7(a;^-3)  =  30. 

17.  6(a;^  +  l)='  +  13(a;'4-l)=-28. 

18.  2l/2^^="3  +  5l^2a;-3  =  7. 


Solve 

1— 

1. 

%'- 

17x^  +  16=0. 

2. 

4a:*- 

-13a;^  +  9  =  0. 

3. 

27a:« 

=  35a;'-8. 

4. 

3.*- 

-5a;i  =  2. 

5. 

27a;' 4-19x^  =  8, 

6. 

3a;^  = 

=  4a;^  +4. 

7. 

2V^ 

=  \/x-\-\. 

8. 

3a;" 

S-f5a;~*  =  2. 

9. 

6x-' 

^-x"*  =  12. 

QUADRATIC  EQUATIONS.  269 

-d- !)■-(! -!)-■ 

20.  5(4x  +  1)  -  27  VWTT=  -  10. 

21.  3(3x^ -  2x  +  1)  - 4 l/3x^-2a;  +  l  =  15. 

22.  2(2x^  +  3x  -  4)^  -  Z(2x'  +  3x  -  4)  =  - 1. 

23.  a^  +  lx -3l/x-'  +  7a;H-l  =  17. 

24.  6(a:'  +  a;)  -  7l/3x(a;  +  1) -2  =  8. 

25.  3x^-7  +  3  l/3^^^=T6^T^r=  16a;. 

26.  32;~^-7x*-4.  28.  16x^-22  =  33;** 

27.  3x^  =  8x~*-10.  29.  2x''-l/x^-2a;-3  =  4a;  +  9. 

30.  5(2x'  -  l)i  -  4  =  f  (2a;'^  - 1)  -  ^ 

31.  3(x»  +  1)  ~  ^ -It  5  =  2(a;' +  l)i 

KADICAL  EQUATIONS. 

254.  Radical  Equations  resulting  in  Affected  Quad- 
ratic Equations.  If  an  equation  be  cleared  of  radicals  by 
the  methods  given  in  Art.  235,  the  result  is  often  a  quadratic 
equation. 

Ex.  Solve  VWYVJ^  VxT2'-=  VHdx  +  16. 


Squaring,     3a;  +  10  +  2l/(3a;  +  10)  (a;  +  2)  +  a;  +  2  =  10a;  +  16 
Hence,  V{Zx  +  lGY{x~^T)  =  3a;  +  2 

Squaring  again,  Sa;^  +  16a;  +  20  =  ^x^  +  12a;  +  4 

6a;^  -  4a;  =  16 

a;  =  2,     -f 

Substituting  these  values  in  the  original  equation,  the  only  value  that 
verifies  is  a;  =  2,  which  is  the  root.  The  other  value,  a;  =  —  f ,  is  not  a 
root  of  the  original  ecjuation,  but  is  introduced  by  sijuaring  in  the  pro- 
cess of  clearing  the  equation  of  radical  signs.     It  satisfies  the  equation, 

^. -.A  ,.         yzxTiQ  -  yirr^  -  vi^x  +  le. 


270  ALGEBRA. 

EXERCISE   96. 

Solve — 


1.  x-l--=  VSx  -  5:  4.  3x  -  2l/Bi  =  6. 

2.  2a;  +  1  =  VTx  +  2.  5.  V^SxTT -  2 l/2x  =  -  3. 

3.  a;  -  l/S^  =  6.  6.  2  +  l/2xTT=  VbxTT. 


7.  1/30;  +  7  =  V^Tl  +  2l/ic  -  2. 

8.  VWTl  =  2Vx-  Vx^^^W. 

9.  1/^'^=^  +  Vx  +  2d'  =  j/x~T7a\ 

10.  1/a;  +  2a  +  2  Ka;  —  2a  =  3a  -  1. 

11.  3l/ar'  +  17-2F'5r'  +  41  +  l/^n=0. 

12.  V'2x  +  3-|l/r=^  =  il/llx-33. 

13.  l/xT4  +  VWTl  -  Vdx  +  4  =  0. 

4a;  +  1 

14.  2l/5x-  t/2x-1 


15 


l/2a;-l 
31/2^-5       9-21/^ 


3+1/25         T/25-3 

16.  VxTJ-^  V^x  —  i  =  VdxTT- 

17.  V^T5"+  l/3x  +  4  —  l/12a:  +  l=0. 

18.  l/12x'^  -  a;  -  6  +  l/12x^  +  a;-6=  l/24a;"^  - 12. 


x+vV^a'      a;- l/a;-^-a''      ^/_,^ ^ 

19.  -_ ^-^=:^  =  SV^^a\ 

X  -  Vx"^^^      x+Vx'-  a' 


20.  l/4x  4-  3  +  V2x  +  3  =-  V5x  +  1  +  1/^+3. 

OTHER    METHODS    OP    SOLVING    QUADRATIC 
EQUATIONS. 

255.  I.  Completing  the  Square  when  the  CoelBficient 
of  ac'  is  a  Square  Number  or  can  be  Readily  made  One. 
If  in  a  simplified  quadratic  equation  the  coefficient  of  x^  is  a 


QUADRATIC  EQUATIONS.  271 

square  number,  we  may  readily  complete  the  square  by  di- 
viding the  second  term  by  twice  the  square  root  of  the  first  term, 
and  adding  the  square  of  the  quotient  thus  obtained  to  both  mem- 
bers of  the  equation. 

That  this  process  gives  a  perfect  square  is  readily  seen  from 
the  fact  that 

{ax  +  by  =  aV  +  2a6a;  +  b\ 

Hence,  given  aV  +  2abx,  the  term  6'  with  which  to  complete 
the  square  may  be  obtained  by  dividing  2ahx  by  2aic,  and 
squaring  the  quotient. 

Ex.  1.  Solve  9x'  +  4x  =  5. 

To  complete  the  square,  take  the  square  root  of  9a:'  (that  is,  3a:),  and 
divide  4a:  by  twice  3a: ;  this  gives  as  a  quotient  f .  Add  the  square  of  this, 
%,  to  both  members. 

.  • .  9a;2  +  4a:  +  I  =  ^. 
Whence  3a;  +  f  =  =t  | 

'  3a;  =  I,     or  -  3 

a;  =  f ,    or  —  1,  Roots. 

Ex.2.   Solve  8a;' +  3a;  =  26. 

To  make  the  coefficient  of  x^  a  square  number,  multiply  both  members  of 
the  equation  by  2. 

.  • .  16a:2  +  6a:  =  52 
Completing  the  square,   ISa:'  +  6a:  +  ^  =  52  +  y*^  =  -^^ 

4a:  +  I  =  ±  ^^ 

X  =  -V->    "~  2,  Roots. 

256.  II.  Hindoo  Method  to  Avoid  Fractions  in  Com- 
pleting the  Square.  After  simplifying  the  equation,  multiply 
through  by  four  times  the  coefficient  of  ac!^,  and  add  to  both  sides  the 
square  of  the  coefficient  of  ac  in  the  simplified  equation. 

The  reason  for  this  process  is  evident,  since  if  aa^  +  bz=^o 
be  multiplied  by  4a,  we  obtain 

4aV  +  4a6a;  =  4ac. 


272  ALGEBRA. 

The  addition  of  h"^  gives  on  the  left-hand  side  4aV  +  Aabx 
+  6',  which  is  a  perfect  square. 

Ex.  Solve  Sx'  -  2x  =  8  by  the  Hindoo  method. 

Multiply  by  4  X  3,     or  12, 

362:2  _  24a;  .=  96. 

Add  the  square  of  the  coeflScient  of  x  in  the  original  equation ;  that  is, 
(-2)2,  or  4. 

36a;2  -  24a;  +  4  =  100 
62:  -  2  =  ±  10 
6x  =12,     -  8 
X  --=  2,     -  I,  Roots. 

EXERCISE   97. 

Solve — 

1.  x'  +  Bx  =  6.     .  11.  (x'  +  3)'  - 1{^  +  3)  =  60. 

2.  3x'-a;  =  2.  12.  4a:-'^- lOla:"' +  25  =  0. 

3.  6x^  +  52;  =  4.  13.  6a:^-5a;^-6. 

4.  1x^  +  11a:  =  6.  14.  4a:^  +  4a:^  =  3. 

6.  8a:'^-2x  =  3.  15.  6  v'^^^  -  1 1  i^^r^  =  10. 

6.  4a:'  +  4a:  -  35.  16.  3(a:  -  2)=^  +  5(a:  -  2)  =  12. 

7.  9x'  -  3a:  =  30.  .  _1  :^      1  JL 

8.  16a:* -40x^  +  9  =  0.  *^      2a:  ~^      2a 

9.  6a:'  -ax  =  2a\  18.  (a  -  l)x'  +  (a  +  l)a:  =  -  2. 
10.  4aV  +  5aa:  =  21.  19.  (a'  -  6')x'  +  (a'  +  i^x  =  aft. 

20.  8a^a:'  -  (66'  +  4a')a:  =  -  Zah. 

257.     III.  Use  of  Formula.     Any  quadratic  equation  can 
be  reduced  to  the  form 

ax'  -\-hx-\-  C—-0. 
Solving  this  equation  by  use  of  Art.  248, 

^  = 2a "• 


QUADRATIC  EQUATIONS.  273 

By  substituting  in  this  result,  as  a  formula,  the  values  of  a, 
5,  c  in  any  given  equation,  the  values  of  x  may  be  at  once 
obtained. 

Ex.  Solve  6x'  +  3a;  —  2  =  0  by  use  of  the  formula. 

Here  a  =  5,     6  =  3,     c  =  —  2. 

Substituting  for  a,  6,  c  in  the  above  formula, 
^       -  3  ±  1/9  +  40 


10 


10 

~  ~  1 

^,     -  1,  Boots. 

EXERCISE  98. 

Solve  by  the  formula — 

1.  2x^  +  bx  =  l. 

9.  2a;' +  ax  =  6a*. 

2.  4a:^-3x  =  7. 

10.  126'  =  3aV-5a6a;. 

3.  6x'  H-  7a;  =  10. 

11.  aa;'  =  (H-a>-a. 

4.  4a;'' -11a;  =  3. 

12.  2a;* -3x4  =  9. 

6.  12x^  +  8a;-15=0.  ' 

13.  6a;-7T/x-20. 

6.  6a;^  +  13a;  +  6=0. 

14.  8x-^  +  19x"'^  =  27. 

7.   33a;'' -  17a;  -  36. 

15.  4x-*-73x-^  =  -144. 

8.  12a;^-a;  =  6. 

16.  3(a;'-l)'  =  7(x'-l)  +  6. 

17.  1(23-  — 3)  — 

I- 

-2)'  =  K18-5x). 

1ft     ^    r 

a; 

l1            ^       . 

18.  — : —    - 

^  +  d 

J     {c-dy 

19       2^  +  ^ 

7- 

■X        7-3x 

•2(2a;-l) 

2a;  +  2       4  -  3a; 

20.  (a  +  3)V- 

(a^- 

-  9)x  =  3a. 

Find  the  approximate  numerical  values  of  x  to  four  deci- 
mal places. 

21.  x'-4x  +  l=0.  23.  2x'  +  3  =  10a;. 

22.  9x'-12a;=-l.  24.  5x'  +  2a;=2. 

18 


274  ALGEBRA. 

EXERCISE   99. 

REVIEW. 
Solve — 

1.  Gx^  +  X  =  1.  9x+-  =  -+-' 

2.  Sx""  +  -\^-x  =  -2/.  b      X      a 

2. 


10.  V4x  -  3  =  1  +  1/a;  + 


4.  a:»  -  16a;  =  0.  n.  Sx     ^  -  7a:     3- 


5.  SVx  -\2x     2  =  5. 


12.  a;*  -  27a:  =  0. 


^a:-l_^      x          oi  13.  5a:-i  +  6a:     ^  =  11 

0.    + —  Zg. 

X         x-\  ^^  x-Z  _  x±_Z  ^  6. 

7       3      _^  2a:  +  1  _i,  'a:-4a;  +  4^* 

*  a:  -  7          3a:          ^  *  15.  2a:2  +  2a:  +  1  =  0. 


8. 


a:4-  1  _  g  +  1  16.  5(a:  +  2)*  -  3(a:  +  2)^  +  2. 

T/a:         Vol  17.  ZVxTJ  -  5l^x~+l.  =  2. 

18.3-(l/^ L\=l/^ 1 — 

^  Vx'  Vx 

19.  20-2(a:  +  ^y  =  3(a:  +  ^)- 

2Q      abx^     +  1  =  (Q^'  +  ^'^)^  . 
•  a2_52  a2-62 

21.  -^4-^=1. 

VX^         1^3 

22.  ^^ =  1+1+1. 

X  -{■  a  +  b       X      a      b 

23   31/^  -  41/2  _   21/^  -  T/g' . 
'  4Vx  -  21/2      3l/3a:  -  51^6 
24.  4x'  -71/2x2  +  3a;_  2  =  19  -  6a:. 


25    1  ^4a:-3 


a;l      3  4      /      '  la:  +  1       a:  -  1  /       '^ 


26.  (x^  -  5xY  -  8(a:2  -  5a:)  =  84.  30.  Sy^  =  36  +  41/^'  _  7^ 

27.  (a;2  +  6a:)"  ~  2(a:2  +  6a:)  =  35.  31.  104  +  UV^  =  Si/^. 

28.  |n2  =  10  -  ^n.  32.  p'^  +  a^  +  y  =  lay  +  a  +  6. 

29.  9n*  =  23n2  +  12.  33.  a:^  -  1  =  (1  -  a:)l/2  -  4a;. 


QUADRATIC  EQUATIONS.  276 

EXERCISE  100. 

1.  Find  two  consecutive  numbers  the  sum  of  whose  squares 
is  61. 

2.  There  are  two  consecutive  numbers,  such  that  if  the 
larger  be  added  to  the  square  of  the  less  the  sum  will  be 
57.     Find  the  numbers. 

3.  There  are  two  numbers  whose  difference  is  3,  and  if 
twice  the  square  of  the  larger  be  added  to  3  times  the 
smaller,  the  sum  is  56.     Find  them. 

4.  Seven  times  a  certain  number  is  one  less  than  the  square 
of  the  number  next  larger  than  the  original  number.  Find 
the  number. 

5.  A  gentleman  is  3  years  older  than  his  brother,  but  twice 
the  product  of  their  ages  is  17  more  than  21  times  the  sum 
of  their  ages.     How  old  is  he? 

6.  If  a  train  had  traveled  6  miles  an  hour  faster,  it  would 
have  required  1  hour  less  to  run  180  miles.  How  fast  did 
it  travel  ? 

7.  Two  numbers  when  added  produce  5.7,  and  when  multi- 
plied produce  8.     What  are  they  ? 

8.  What  are  the  two  parts  of  18  whose  product  exceeds  8 
times  their  difference  by  1? 

9.  A  gentleman  distributed  among  some  boys  $9;  if  he 
had  begun  by  giving  each  boy  5  cents  more,  6  of  them 
would  have  received  nothing.  How  many  boys  were 
there? 

10.  A  cistern  is  filled  by  two  pipes  in  18  minutes ;  by  the 
greater  alone  it  can  be  filled  in  15  minutes  less  than  by  the 
smaller.     Find  the  time  required  to  fill  it  by  each. 

11.  A  certam  number  of  eggs  cost  a  dollar,  but  if  there 
had  been  10  more  eggs  at  the  same  price,  they  would  have 
cost  6  cents  a  dozen  less.  What  was  the  price  of  a  dozen 
eggs? 

12.  One  number  is  |  of  another,  and  their  product,  plus 
their  sum,  is  69.    Find  the  numbers. 


276  ALGEBRA. 

13.  Find  two  numbers  whose  product  is  90  and  quo- 
tient 21. 

14.  Find  two  numbers  whose  difference  is  4  and  the  sum 
of  whose  squares  is  170. 

15.  Find  two  numbers  whose  product  is  42,  such  that  if 

the  larger  be  divided  by  the  less,  the  quotient  is  4  and  the 

remainder  2. 

49 
[Let  X  and  —  represent  the  numbers.] 

16.  A  number  of  boys  bought  a  boat,  each  paying  as  many 
dollars  as  there  were  boys  in  the  party;  had  there  been  5 
boys  more,  and  each  paid  f  as  much  as  he  did  pay,  they 
would  have  lacked  $10  of  the  price  of  the  boat.  How  many 
boys  were  there  ? 

17.  A  company  of  gentlemen  agreed  to  buy  a  boat  for 
$7200,  but  3  of  their  number  died,  and  each  survivor  was 
obliged  to  contribute  $400  more  than  he  otherwise  would 
have  done.     How  many  men  were  there? 

18.  Divide  the  number  12  into  two  parts,  such  that  the 
sum  of  the  fractions  obtained  by  dividing  12  by  the  parts 
shall  be  ff. 

19.  The  length  of  a  certain  rectangle  is  twice  its  width,  and 
it  has  the  same  area  as  another,  1^  times  as  wide,  and  shorter 
by  4^  feet.     Find  its  length. 

20.  A  rectangular  lot  is  8  rods  long  and  6  rods  wide,  and  is 
surrounded  by  a  drive  of  uniform  width  which  occupies  |  as 
much  area  as  the  lot.     Required  the  width  of  the  drive. 

21.  A  rectangular  lot  20  by  15  rods  is  surrounded  by  a 
fence,  within  which  is  a  drive  occupying  as  much  area  as 
the  rest  of  the  lot.     Find  its  width. 

22.  A  number  of  two  figures  has  the  units'  digit  double  the 
tens'  digit,  but  the  product  of  this  number  and  the  one  ob- 
tained by  inverting  the  order  of  the  figures  is  1008.  Find 
the  number. 

23.  A  cistern  can  be  filled  by  2  pipes  in  1  hour  and  33f 


QUADRATIC  EQUATIONS.  277 

oiiniites,  but  the  larger  alone  can  fill  it  in  1  hoar  and  40  min- 
utes less  than  the  smaller  one.  Find  the  time  required  by 
the  less. 

24.  The  left-hand  digit  of  a  certain  number  of  two  figures 
is  f  of  the  right  digit.  If  the  product  of  this  number  and 
the  number  obtained  by  inverting  the  order  of  the  digits  be 
increased  by  twice  the  original  number,  the  sum  is  800. 
Find  the  number. 

25.  A  man  can  row  down  a  stream  16  miles  and  back  in  10 
hours.  If  the  stream  runs  3  miles  an  hour,  find  his  rate  of 
rowing  in  calm  water. 

26.  Two  trains  run  at  uniform  rates  over  the  same  120 
miles  of  rail ;  one  of  them  goes  5  miles  an  hour  faster  than 
the  other,  and  takes  20  minutes  less  time  to  run  this  distance. 
Find  the  rate  of  the  faster  train. 

27.  A  and  B  accomplish  a  certain  task  in  a  certain  time, 
but  if  each  were  to  do  half  the  work,  A  would  work  2^  days 
more,  and  B,  1^  days  less  than  if  they  work  together  till  the 
work  is  completed.  Find  the  time  required  for  each  to 
do  it. 

28.  If  a  carriage  wheel  11  feet  in  circumference  took  -^  of 
a  second  less  to  revolve,  the  rate  of  the  carriage  would  be  1 
mile  more  per  hour.    At  what  rate  is  the  carriage  traveling  ? 

Solve— 

a  +  b  a 


29.  — ^'^—  + 
a  -  X  X  a- X 

30.  2cx2  +  2a\x  +  c)  =  ax{x  +  5c). 

31.  a{h  -  c)x^  +  6(c  -  a)x  +  c{a  -  6)  =  0. 

32.  (4a^  -  962)  ^^.2  +  i)  =  ^xi^a"  +  96'). 

2. 


33. 

a  —  h^-x  ^  a  +  6 
a  +  6  +  a;    'a;  +  6 

34. 

d+  46  a  -  46  _ 
a;  +  26      a;  -  26 

35. 

i^-(-«^> 

at      b 


CHAPTER    XX. 
SIMULTANEOUS  QUADRATIC  EQUATIONS. 

258.  The  General  Problem.  If  the  relations  of  two  un. 
known  numbers  to  known  numbers  be  given  in  the  shape 
of  two  quadratic  equations,  the  problem  is  to  combine  these 
relations  so  as  to  obtain  simpler  ones  which  will  show 
directly  the  values  of  the  unknown  numbers.  This  can  be 
done  in  certain  special  cases  only,  if  we  limit  the  work  to 
methods  already  given  for  solving  quadratic  equations. 

259.  A  Homogeneous  Equation  is  one  in  which  all  the 
terms,  containing  an  unknown  quantity,  are  of  the  same 
degree.     Thus, 

3^V-5^2/'  +  /  =  18 
is  homogeneous,  and  of  the  third  degree. 

CASE  I. 

260.  When  One  Equation  is  of  the  First  Degree,  the 
Other  of  the  Second. 

Two  simultaneous  equations  of  the  kind  just  specified 
may  always  he  solved  hy  the  method  of  substitution, 

^      ^  ,  (2x-Sy  =  2 (1) 

Ex.  Solve  )  J  V  y 


lx'-2xy=-7 (2) 

From(l),  2/  =  ^^ (3) 

Substitute  for  y  in  (2), 


^     2x-2\  „ 


Hence,  Sx'  -Ax'  +  4x  =  -  21 

ic'-4a;  =  21 


Substitute  for  x  in  (3), 

278 


SIMULTANEOUS  QUADRATIC  EQUATIONS  279 

It  is  to  be  observed  thnt  corresponding  values  of  x  and  y 
must  be  used  together.  Thus,  when  x=  —^,y  must  =  —  f, 
and  not  4.     Likewise  the  values,  7  and  4,  go  together. 


- 

EXERCISE  101. 

Find  the  values  of  x  and  y- 

- 

1.  3a;^-22/^  =  -5. 

8.  io;  -  iy  =  i. 

a;  -|.  2/  -  3  =  0. 

{.x-yy  =  y--l. 

2.  x  —  2y  =  S. 

9.  x'-3x2/  +  22/'=0. 

x'  +  4y'  =  17, 

2x  +  32/  =  7. 

3.  2x'-\-xy  =  2. 

10.  42/'  +  42/  =  4x-13. 

3a; +  2/ =  3. 

.     .           102/-2X-1. 

4.  rc^-32/^  =  l. 

11.  9^  —  62/  — 5  =  3ar. 

a:  +  22/  =  4. 

92/  +  x  +  5=0. 

6.  x-Sy  =  l. 
lxy-x^  =  12. 

6.  2a;  +  2/H-3-0. 
3x^-72/^  =  5. 

7.  2a;  +  52/  =  l. 

3      2       9 
12.  ---  =  — 

y      X     xy 

2x      10      32/     ^ 
1 —  =5. 

2/       X2/       X 

13.^-^-^  =  1. 

2x      32/ 

2x^  +  3x2/ =  9. 

3:y  +  4x  =  6. 

14.  3x- 

-by- 

1=0. 

2x'  +  3x2/ 

-52/'-6x  +  72/  =  4. 

15.  4x^- 

-4x2/ 

=  2/^  +  x  +  32/-l. 

4x 

-2- 

52/  =  0. 

CASE   II. 

261.  When  both  Equations  are  Homogeneous  and  of 
the  Second  Degree. 

Two  simultaneous  quadratic  equations  of  this  kind  can 
always  de  solved  by  the  substitution  y  —  vx. 


280  ALGEBRA. 

Ex.  Solvfe  x'-xy  +  y' =  21, 

2/'  —  2xy  =  —  15. 

Substitute  y  =  vx,  x^  —  vx'^  +  v^x^  =  21 (1) 

v^x'  -  2vx^  =  -  15 (2) 

From(l),  -'-i^f^ (^) 

From(2),  ^.  =  _=i|_  . (4) 

Equate  the  values  of  x^  in  (3)  and  (4), 

21  _      -15 

1  -  V  -\-  v'^       v^  —  2v 
Hence,  21^2  -  42i;  =  -  15  +  15v  -  15i;» 

36t;2  -  57v  =  -  15 

12v2  -  19i;  -  -  5  .  • .  V  =  f,  I 

Substitute  for  v  its  values  in  (3), 

^2  ^  21  21__ 

^  1    _    5    4.    25'  ^^     1    _    1    +    1 

Hence,  a;  =  =t  4,     or  ±  31^3 

Since  y  =  vx,  multiply  each  value  of  x  by  the  corresponding  value  of  V, 

.•.y=(±4)|=  i5, 

y  =  (i3l/3)^=  il/3". 


EXERCISE  102. 

Find  the  values  of  x  and  2/ — 

1.  x"  +  82:2/  =  28. 

5.  2x'-f  =  ie, 

xy-\-^f  =  ^. 

^y  +  y'  =  14. 

2.  2x'  +  a:2/  =  15. 

6.  3x'  +  f  =  12. 

x'-y'  =  S. 

5xy-4x':=ll. 

3.  x'  +  3a;2/=7. 

7.  22/' -  4x2/ -f  3x^  =  17. 

y^-\-xy  =  6. 

2/'-x'-16. 

4.  2x^-32/^  =  6. 

8.  x'  +  xy  +  2f  =  74. 

3x2/ -42/' =  2. 

2x^  +  2x2/ +  2/' =  73. 

SIMULTANEOUS  QUADRATIC  EQUATIONS.          281 

9.  22;'  +  Sxy  +  y'  =  14.  11.  x'  +  xyi-  2y'  =  U, 
'6x'  +  2xy-4y'  =  d.  2x'' -  xy  +  y' =  IQ. 

10.  4xy-x'  =  5.  12.  2x^  -  7xy  -  2y' =  5, 
13x'  -  Slxy  +  Uy'  =  2^.  Sxy  -  x'  +  Qy'  -  44. 

SPECIAL  METHODS  OP  SOLVING  SIMULTANEOUS 
QUADRATICS. 

262.  The  methods  of  Cases  I.  and  II.  are  the  only  genera] 
methods  which  can  be  used  in  solving  all  simultaneous  quad- 
ratic equations  of  a  given  class.  Besides  these,  however,  there 
are  certain  special  methods  which  enable  us  to  solve  import- 
ant particular  examples. 

Examples  which  come  directly  under  Cases  I.  and  II.  are 
often  solved  more  advantageously  by  one  of  these  special 
methods. 

The  special  methods  apply  with  particular  advantage  to 
what  are  called  symmetrical  equations. 

263.  A  Symmetrical  Equation  is  one  in  which,  if  y  be 
substituted  for  x,  and  x  for  y,  the  resulting  equation  is  iden- 
tical with  the  original  equation. 

Thus,  each  of  the  following  is  a  symmetrical  equation : 

X  -\-  y  =  12,  xy  =  &. 

264.  I.  Addition  and  Subtraction  Method  (often  in  con- 
nection with  multiplication  and  division).  In  this  method 
the  object  is  to  find,  first,  the  values  ofoc-^y,  and  00  -  y,  and 
tlien  the  values  of  x  and  y  themselves. 

,  _  ,      ,     .   , (1) 

Ex.1.  Solve 


[x  +  y  =  l 
I      xy  =  V. 


xy  =  12 (2) 

Here  we  have  the  value  of  x  +  y  given,  and  the  object  is  to  find  the 
value  oi  X  —  y. 

Square  (1),  a;»  +  2a;y  +  y'  =  49 (3) 

Multiply  (2)  by  4,  4a^  =  48 (4) 


282 


ALGEBRA. 


Subtract  (4)  from  (3),  x 

Extract  square  root  of  (5), 
Add{l)  and  (6),  divide  by  2, 
Subtract  (6)  from  (1),  divide  by  2, 


2  -  2xy  4-  2/^  =  1  . 
x-y  =^  =t  1 


rt'=  4  or  3 


y 


4  or  3  V 
3  or  4J 


Boots. 


Ex.  2.   Solve 


Divide  (1)  by  (2),  x"  -  xy  ^-  y"  =  13 

Square  (2),  x"  +  Ixy  +  ^/^  =  25 

Subtract  (3)  from  (4),  Zxy  =  12 

Hence,  xy  ==  ^  . 

Subtract  (5)  from  (3),     x^  -  2xy  +  y"^  =  ^ 

.♦.  x-y=  ±3 


But 
Hence, 


Ex.3.  Solve 


a;  +  2/  =  5 


(1  +  ^=11. 

I  ^      y 


Boots, 


Squaring  (1), 
Subtracting  (2)  from  (3), 


x'       2/ 
+  —  +  A  =  121 


Subtracting  (4)  from  (2),  -I  -  A  +  1  =  i 


Hence, 

But,  from  (1), 

Hence,  adding, 


c^  xy  y' 
2^ 
xy 

1 

xy      y 

1 

y 
J_ 
y 


—  =  60 


X 


i  +  l-u 


t  -  12,  10 

3^  =  i  \] 


Boots, 


(5) 

(6) 


(1) 

(2) 
(3) 
(4) 

(5) 


(1) 

(2) 

(3) 
(4) 
(5) 


SIMULTANEOUS   QUADRATIC  EQUATIONS  283 

EXERCISE  103. 

Find  the  values  of  x  and  y — 

I.  x^y  =  U.  X3.  a^  +  2/»  =  2a'  +  6a. 

^.y  =  26.  ic'-x?/  +  2/'^a'  +  3. 

x  +  y  =  l,  y      ^        ^ 

S,x  +  y=~10.  x  +  y  =  5. 

^y  =  '21,  15.  ar*  +  2/' =- 224. 

4.  a:^  +  a:?/ -f  2/^^ --- 21.  a:^  +  a:^/' =  96. 
a;  +  2/==  -  1.  11 

5.  x'-xyi-y^  =  S7.  ^^'  ?  "^  7  "  ^^* 
x^  +  xy  +  y^  =  79,  l_fi=o 

6.  x'  +  2/^  =  2i-.  2:2/ 
3xv  =  2i. 

7.  a:  +  2/  +  l=0.  ^         ^ 

^2/  +  3i=^0.              .  17.  -  +  -=3}. 


8.  x^  +  'if  =  d. 


^     y" 


x  +  y^S.  1  +  1  =  2 

9.  a;'  +  2/'  =  37.  x      y        ' 
x  +  y  =  l. 

10.  x3  + 2/^-218  ==0.  18.  x»  +  2/'--J:«^. 
a:'  -  X2/  +  7/  -  109.  _ 

11.  x^+Sxy+y^=  — 2f .  «  +  2/  =-  i 
x''-a:2/  +  2/'  =  12J. 

12.  xy-Qd^  =  0.  1^-  a:*  +  a:y  +  y  =  4^. 

a:^  +  2/'  =  a.-^  +  7a\  x'  +  a;2,'  +  y'  =  lh 
Solve  also  by  the  same  method — 

20.  x'  +  y'  =  ^.  22.  x'  +  2/'  ^  5(a'  +  b'). 
x  —  y  =  A.  y-x  =  a  +  Zh, 

21.  a:'  -  2/'  =-  98.  23.  Z^  +  Saty  +  82/'  -  13. 
x-y  =  2.  5x^  +  3x2/ +  52/' =  27. 


284  ALGEBRA. 

265.     II.  Solution  by  the  Substitutions,  x  =  u  +  v  and 

y  =  u  —  v, 

[0:^  +  2/^  =  242 (1) 

Ex.  Solve  1       ,         o  ro\ 

Substitute  in  (1)  and  (2),  x  =  u  +  v,    y  =  u-v 

From  (1),  2w*  +  20^3^2  +  IOmv*  =  242 (3) 

From  (2),  2w  =  2 (4) 

Divide  (3)  and  (4)  by  2,  and  substitute  in  (3)  for  u,  i.  e.,  w  =  1, 

1  +  10y2  +  5?;*  =  121 
Hence,  v^  +  2v'' =  24 

V  =  ±  2,    ±  1/^=^ 
But  w  =  1 

Hence,  a;  =  w  +  v  -  3,   -  1,  1  ±  l/^6|  ^^^^ 

y  =  w-t;=-l,  3,  1  ^\^^Q) 


266.  III.  Use  of  Compound  Unknown  Quantities.  It 
is  often  expedient  to  consider  some  expression,  as  the  sum, 
difference^  or  product  of  the  unknown  qua7itities,  as  a  single 
unhnoiun  quantity,  and  find  its  value^  and  hence  the  value  of 
the  unJcnown  quantities  themselves. 

Ex.  Solve  i.^-Vf  =  iy.-y (1) 

I        xy  =  Q>     (2) 

Add  2xy  =  12  to  (1). 

Then.  x"^  +  "tcy  -\-  y"^  =^ '^0  -  x  -  y (3) 

Let  X  +  y  =  V, 

Then  from   (3),  v^  =  30  -  v 

^2  +  V  =  30 

V  =  -  6,  5 


Hence,         x  -^^  y  =  —  % 
xy  =  6. 
,'.x=  -3  ±1/3, 

y  =    _  3   :p  1/3. 


also    x  +  y  =  5, 
xy  =  6. 
.'.  a;  =  3,  2, 
y  =  2,  3. 


SIMULTANEOUS    QUADRATIC  EQUATIONS.  205 

EXERCISE   103  (A). 

Find  the  values  of  x  and  y. 

1.  x5  +  7/5=:244;  x-f  t/  =  4.  x  =  3,  1,2±3  i/^T7  y  =  l,  3,  2=F3  ^^X 

2.  a;2-ft/2  4.x4-y  =  24;  a;t/  =  -]2.  a;  =  3,  -  4,  ib2i/3';  y=-4,3,=F2  v/ST 

3.  x-]-y-\-^x-\-y  =  Q;  xy=3.        x  =  l,3, 1(9  ±^69);  y  =  3,l,^{9  =f  ^/W). 

4.  xV  +  a:y  =  6;a:  +  2y  =  -5.  «  =  1,  -  6,  -  4,  -  1;  l/  = -3,^-^-2. 

5.  a- 4-^  =  25;  -/"E-f  i/ y  =  a;  -  y.  x  =  9,  16;  t/ =  16,  9. 

6.  y  +  i/a;-''- 9  =  6;  i/x  +  3  -  •i/5"^=3=  i/y.  a:  =  3,  5;  y  =  6,  2. 

7.  a:*  +  2/*  =  y7.  10.*  {x  —  yy—Z{x—y)  =40, 

8.  a:'^  +  2/2  =  a:-y+50.  ^l'  a^' +  2/' +  a:  +  5t/ =  6. 

9.  xY  +  7xy  =  -  6.  12.t 'o:^  y^  {x^  +  y^)=  70. 
5a;2  +  a:y  =  4.  a^i/^  4_ a;i  4.  y^=  I7. 

EXERCISE  104. 

GENERAL  EXERCISE. 

Find  the  values  of  x  and  y — 

1.  2x-5y  =  0.  6.  a^  +  y*  =  n. 

6.  a;'  +  32/'  =  28. 
2-^  +  2/  =  2.  ^  +  xy  +  2y^  =  lQ. 

?_L?=6  7,xy  +  2x  =  5. 

a?      2/        '  2xy  —  y=-S. 

3.  2a;'-a;2/  =  28.  8.  Zx" -^  xy -^  y' =  16 


4x      22/_34 
3a;-62/  =  l.  2a:-52/=-4. 


*  There  are  eight  roots  for  x  and  eight  for  y. 
t  Coj^^ider^  first,  tb»<t  the  unknown  quantities  are 
^  =  i/xy  and  v  =  i/*  +  i/y7 


286 


ALGEBRA. 


10.  x'  +  2x  —  y  =  5. 
2x'-Bx'\-2y  =  8. 

11.  (x  +  2)(2y-l)  =  B5, 
xy  —  x  —  y-=7. 

12.  xY-6xy  +  6  =  0, 
6x  +  Sy  =  14. 

13.  (x  +  yy-ix-hy)  =  20. 
2x'  ~Sx-{-4y  =  14. 

14.  x'  +  y'  +  x  +  Sy^r-lS. 
xy-y  =  12. 

15.  x^  —  'j/=—  Zxy, 
x-y  =  2. 

16.  — +  — =f. 
y        X 

1  1  AX 

X      y 

17.  x-^  +  «/-^  =  2. 
^-'  +  y-'  =  2i. 

18.  a:'  +  2/'  +  a;  +  2/  =  14. 
xy  -\-x-\-y=  —  5. 

19.  3x'  -  35  =  5x2/  -  72/'. 
2a;2  — 35=-Y  — X2/. 

20.  a;* +  2/*  =  17. 
a;  +  2/  ==  3. 

21.  x'  +  2/'  =  211. 

a;  +  2/  =  l- 

22.  x*  +  2/*  =  82. 
a;  +  2/  =  2. 

23.  x*  +  2/*  =  257. 
«-2/  =  5. 


24.  a;*  +  2/*  =  17. 
a;2/  =  2. 

25.  x'  4-  2/"^  =  ^2/  +  7. 
a;  —  2/  —  ^2/  —  5. 

26.  x' +  2/' =  a;2/ +  13. 
X  +  2/  =  XT/  —  5. 

27.  x^  =  4(a^  +  6^-2/0. 
X2/  =  2ab. 

1      1      , 

28.  -  +  -  -:  5. 
X      y 

x  +  1       2/  +  1       ''' 

oq    a;-2/       a;-4-2/_, 

^y.  —  6". 

x  +  2/       X  — 2/ 

2x  +  52/  =  5. 

80.  3x  — 42/  =  a. 

2x'^  -  32/'  +  a2/  =  8al 

31.  X  — 4  =  2/(^-2). 
2/-8  =  x(2/-2). 

32.  2x-^  +  5?y-i  =  4. 

33.  x2/  +  a;  +  2/  =  5. 
x^2/  +  X2/^  =  —  84. 

34.  ax  +  hy  =  0. 
(ax-2)(62/  +  3)--2. 

35.  ^  +  ^=3J. 
2a      36        ' 

4a      56       . 

_  +  — =4. 
a      y 


SIMULTANEOUS  QUADRATIC  EQUATIONS.  287 

36.  a(x  —  a)-=h{y  —  h).  40.  3!?-\-x  =  ^y. 
xy=-ax-\-  by.  x'  +  l^Qy. 

37.  x'  +  ay  =  ii-a'x'y\  41.  x' ^  f  =  Sxy  -  4. 

Sx  +  ay  =  5.  x*  +  y*  ^  272. 

38.  x-\-y  =  65.  42.  a;  +  V^  +  y  -=  14. 
V'S  +  v^^  =  5.  x'  +  a;2/  4-  2/'  =  84. 

39.  x~y^  Vx+  Vy.  43.  a:^  +  4^/^  +  80  =  16x  +  30y, 
^f_2/l^37.  a:2/-6=0. 


EXERCISE  105. 

1.  The  sum  of  the  squares  of  two  numbers  is  58,  and  their 
product  is  21.     Find  the  numbers. 

2.  Find  two  numbers  whose  sum  increased  by  three  times 
their  product  is  83,  and  of  which  3  times  the  less  exceeds  the 
larger  by  1. 

3.  The  area  of  a  rectangle  is  84  square  feet,  and  the  dis- 
tance around  it  (perimeter)  is  38  feet.  Find  the  length  and 
breadth  (dimensions)  of  the  rectangle. 

4.  If  the  dimensions  of  a  rectangular  field  were  each  in- 
creased by  3  rods,  its  area  would  be  140  sq.  rds. ;  but  if 
its  width  were  increased  by  8  rods  and  length  diminished 
by  2,  its  area  would  be  135  sq.  rds.  Find  its  actual  dimen- 
sions. 

5.  A  rectangular  lot  containing  270  square  rods  is  sur- 
rounded by  a  road  1  rod  wide ;  the  area  of  the  road  is  70 
square  rods.     Find  the  dimensions  of  the  field. 

6.  A  certain  number  of  two  figures  when  multiplied  by  the 
left  digit  becomes  56 ;  but  if  by  the  right  digit,  224.  Required 
the  number. 

7.  A  hall  of  90  square  yards  can  be  paved  with  720  rect- 
angular tiles  of  a  certain  size,  but  if  each  tile  were  3  inches 
shorter  and  3  inches  wider,  it  would  require  648  tiles.  What 
is  the  size  of  each  tile  ? 


^88  ALGEBRA. 

8.  A  merchant  bought  a  number  of  yards  of  cloth  for  $140 ; 
he  kept  8  yards  and  sold  the  remainder  at  an  advance  of  $1^ 
a  yard,  and  gained  $20.     How  many  yards  did  he  buy  ? 

9.  Two  farmers,  A  and  B,  have  together  30  calves,  which 
they  sell  for  $336,  A  receiving  as  many  dollars  for  each  of  his  as 
B  had  calves ;  if  they  had  each  sold  his  calves  for  as  many  dol- 
lars apiece  as  the  other  received  for  each  of  his,  they  would 
have  received  only  $324.  How  many  calves  had  A,  and  at 
what  price  did  he  sell  them? 

10.  The  sum  of  the  numerator  and  denominator  of  a  cer- 
tain fraction  is  8,  and  if  2-^  be  added  to  each  term  of  the  frac- 
tion, its  value  will  be  increased  by  -^.     What  is  the  fraction? 

11.  Two  trains  traveling  toward  each  other  left,  at  the  >same 
time,  two  stations  240  miles  apart ;  each  reached  the  station 
from  which  the  other  started,  the  one  3f  hours,  and  the  other 
If  hours,  after  they  met.     Required  their  rates  of  running. 

12.  A  crew  rowing  at  f  their  usual  rate  took  32  bours  to 
row  down  stream  48  miles  and  back  to  starting-place ;  had 
they  rowed  at  their  usual  rate  it  would  have  taken  18  hours 
for  same  circuit.     Find  their  rate  and  that  of  the  stream. 

13.  Two  square  plots  contain  together  610  square  feet,  but 
a  third  plot,  which  is  a  foot  shorter  than  a  side  of  the  larger 
square,  and  a  foot  wider  than  the  less,  contains  280  square 
feet.    What  are  the  sides  of  the  two  squares  ? 

14.  The  fore  wheel  of  a  carriage  makes  28  revolutions  more 
than  the  hind  wheel  in  going  560  yards,  but  if  the  circumfer- 
ence of  each  wheel  were  increased  by  2  feet,  the  difference 
would  be  only  20  revolutions.  What  is  the  circumference 
of  each  wheel? 

15.  A  number  of  foot-balls  cost  $100,  but  if  they  had  cost 
$1  apiece  less,  I  should  have  had  as  many  more  for  the 
money  as  the  number  of  dollars  paid  for  each  ball.  Find 
the  cost  of  each. 

16.  Find  two  fractions  whose  sum  is  equal  to  their  product 
and  the  difference  of  whose  squares  is  f  of  their  product. 


CHAPTER    XXI. 

GENERAL  PROPERTIES  Of  QUADRATIC  EQUA- 
TIONS. 

267.  Two  General  Forms   of  the  Quadratic  Equation. 
Any  quadratic  equation  may  be  reduced  to  the  general  form 

ax'  +  bx  +  c^O  .......   ^-  I. 

Factoring,  this  becomes 


a(x'  +  -x-\--]  =  0. 
\         a       aj 


h  c 

Dividing  by  a  and  denoting  -  by  p,  and  -  by  g,  we  obtain 

(I  CL 

x''  +  px  +  q  =  0 II. 

When  a,  6,  c,  or  p  and  q  are  given,  we  can  often  infer  at 
once,  without  the  labor  of  solving  the  equation,  important 
facts  concerning  the  roots  of  an  equation.  Or  if,  on  the  other 
hand,  the  roots  only  of  an  equation  be  given,  or  some  prop- 
erty of  them,  we  can  at  once  infer  what  the  equation  will  be. 

PROPERTIES  OIP  x^  +  px  +  q  =  O. 

268.  Relation  of  the  Roots  of  x^+px  +  4  =  0  to  the 
Coefficients  p  and  q. 

Solving  a;^  +_pa;  +  g  =  0,  and  denoting  its  roots  by  n,  ra,  we 
obtain 


p       Vf-4q 
2             2 

p       Vf-4q 
'^""      2              2 

Adding, 

n  +  n  =—P' 

Multiplying, 

rir,  =  q. 

19 

m 


290  ALGEBRA. 

Hence, 

(1)  The  sum  of  the  roots  ofx^  +  px  +  q  —  O  equals  —p^  or  the 
coefficient  of  ac  with  the  sign  changed; 

(2)  The  prqcluct  of  the  roots  equals  the  known  term  q, 

Ex.  In  x'  -  5a;  +  6  =  0 

the  roots  are  found  to  be  3,  2. 

The  sum  of  these  with  the  sign  changed  is  —  5,  the  coeffi- 
cient of  X ;  the  product  is  6,  the  known  term. 

This  relation  is  used  in  the  factorial  method  of  solving 
quadratic  equations    (See  Art.  250.) 

269.  Formation  of  a  Quadratic  Equation,  the  Roots 
Only  being  Given. 

If  the  two  roots  of  a  quadratic  equation  be  given,  the 
equation  may  at  once  be  written  out  by  the  use  of  the  rela- 
tion between  the  roots  and  coefficients  determined  in  Art. 
268. 

Ex.  Form  the  quadratic  equation  whose  roots  are  5, 
and  —  2. 

The  sum  of  5  and  —  2  is  +  3 ;  hence,  the  coefficient  of  x 
in  the  required  equation  is  —3. 

The  product  of  5  and  —  2  is  — 10,  the  third  term ;  hence, 
a;*  —  3a;  —  10  =  0  is  the  required  equation. 

This  equation  might  have  been  formed  also  by  subtracting 
each  root  from  x,  multiplying  together  the  binomials  thus 
formed,  and  letting  the  product  =  0. 

Thus,  (a;-5)(a;  +  2)-0, 

or        a;' -3a; -10  =  0. 

270.  Factoring  a  Quadratic  Expression.  Any  quadratic 
expression  may  be  factored  by  letting  the  given  expression 
equal  zero,  and  using  the  property  stated  in  Art.  268. 


PROPERTIES  OF  QUADRATIC  EQUATIONS.         291 

Ex.  1.   Factor  a;' —  4aj  +  2. 

Solving  the  equation,     a;''  —  4a;  +  2  =  0 

a;  =  2  =tV2 
.•.a:2-4a;  +  2=(a;-2-V2)(a;-2  +  1/2),  Fadon. 

Ex.2.   Factor  Sx'^  -  4x  +  5. 

Take  S{x'  -  fx  +  f )  =  0 

Solve  a;2  -  |a;  +  f  -  0, 

Whence  x=  2=^1/- 11  , 

3 

Hence,  the  factors  of  Sx^  —  4a;  +  5  are 


EXERCISE  106. 

Find  by  inspection  the  sum  and  product  of  the  roots  in 
each  of  the  following  equations: 

1.  x'  +  ^x  +  6=^0.  6.  aV  -  ax  +  2  =  0. 

2.  x'-x-\-7=0.  7.  5x-4x'^  =  l. 

3.  x''-5x  =  10.  8.  S-Tx^llx". 

4.  2a;'-6x-3  =  0.  9.  4x^-ax  +  x=^a\ 

6.  6a;'-a;  =  l.  10.  1  -  2ca;  -  2aa;' =  3«. 

Form  the  equations  whose  roots  are — 

11.  2,  3.  19.  li  -2*.  „     2±l/2 

25. • 

12.  2,  -1.  20.  1  +  a,  l-«.  2 

18.  3,  —2.  21.  a6,  -a.  1 -h  y^ZTl 

14.-1,-5.  ^         ,  26. 


15.  i,  6.  22.  - 


a        b  2 


6        a 


-2±:l/' 


16.  -1,  -i  27. 

17.  -  2,  - 1.  23.  1  +  V^2,  1  -  1/2.  2 

18.  I,  -|.  24.   -3  ±1/3.  28.  iadzcl/^^ 


292  ALGEBRA. 

Factor — 

29.  3a;'  — lOc  — 8.  84.  x'  +  14-6a;. 

30.  24a;' +  2a;  -  15.  35.  25a;' +  2  -  30a;. 

31.  a;'  +  2x  -  1.  36.  4a;'  -  8a;  +  7. 

32.  a;'  —  4a;  +  1.  37.  5a;'  +  6a;  +  7. 

33.  a;"  -  a;  -  1.  38.  3x  -  3a;'  -  1. 

PROPERTIES   OP  ax"  +  boc  +  c  =^  O, 

271.  Character  of  the  Roots  Inferred  from  the  Coef- 
ficients. It  is  important  to  be  able  to  infer  at  once  from  the 
nature  of  the  given  coefficients,  a,  6,  c,  of  an  equation  in  the 
form  ax'  -f  6a;  +  c  =  0,  whether  the  roots  of  the  equation  be 
equal  or  unequal,  real  or  imaginary,  positive  or  negative. 

Solving  ax'  +  6x  +  c  =  0,  and  denoting  the  roots  by  n,  rj,  we 

obtain 

-  6  4-  Vb'-4ac  -b-  Vb^  -  4ac 

ri= 1    ra= 

2a  2a 

From  these  expressions  we  infer  that 

I.  If  b"^  —  Aac  !>  0,  the  roots  are  real  and  unequal. 

For  if  6'  —  4ac  is  a  positive  quantity  (greater  than  zero),  the 
radical  1^5'  —  4ac  is  real  and  not  imaginary,  and  since  the 

fraction  of  which  it  is  a  numerator  is  added  to to  form 

2a 

one  root,  and  subtracted  from ^  to  form  the  other  root, 

2a 

the  two  roots  are  unequal. 

The  roots  are  also  rational  or  irrational,  according  as 
6'  —  4ac  is  or  is  not  a  perfect  square. 

The  roots  are  also  rational  if  6'  —  4ac  =  0. 

II.  if  6'  —  iac  =  0,  the  roots  are  real  and  equal,  since  each 

rootreduceato-l. 
-  2a 


PROPERTIES  OF  QUADRATIC  EQUATIONS.         293 

III.  Jj  h^  —  4:ac  <  0,  the  two  roots  are  imaginary. 

Since  the  character  of  the  roots  is  thus  determined  by  the 
value  of  b'^  —  AaCj  this  expression  is  termed  the  discriminant 

of  ax^  -f-  6x  +  c  ==  0. 

]']x.  1.  Determine  the  character  of  the  roots  of  the  equa- 
tion, 2x"'  +  7x-15=0. 

We  have  a  =  2,    6  =  7,     c  =  —  15. 

.  • .  6'  -  4ac  =  49  +  120  =  169. 
Hence,  the  roots  are  real,  rational,  and  unequal. 

Ex.2.   Of  9a:''-12x  +  4  =  0. 

Here  a  =  9,     6  =  -  12,     c  -  4 

.'.b'-  iac  =  144  -  144  =  0. 

Hence,  the  roots  are  real  and  equal. 

Ex.  3.   Of  3a;'-4a;  +  2  =  0.        ^ 

Here  a  =  3,     6=-4,     c  =  2 

.  • .  6^  -  4ac  =  16  -  24  =  -  8. 
Hence,  the  roots  are  imaginary. 

272.  Determining  Coefficients  so  that  the  Roots  shall 
satisfy  a  Given  Condition.  It  is  often  possible  so  to  deter- 
mine the  coefficients  of  an  equation  that  the  roots  shall  satisfy 
a  given  condition. 

Ex.  Find  the  value  of  m  for  which  the  equation  (m  —  l)x' 
H-  mx  +  2m  —  3  =  0  shall  have  equal  roots. 
By  Art.  271,  II.,  in  order  that  the  roots  be  equal,  b"^  -  ^ac  =  0. 
In  the  given  equation,     a  =  m  -  1,     b  =  m,     c  =  2w  —  3. 
.-.  m2-4(m-  l)(2m-3)  =0 
m^  -  8m^  +  20m  -  12  =  0 

7?7i2  _  20m  =  -  12 
m  -  2,  f . 
Proof.    Substituting  these  values  for  m  in  the  original  equation, 
x'  -j-  2x  t  1  =0,        a;2  -  6a:  f  9  =  0, 
Ol  each  of  which  equations  the  roots  are  equal. 


294  ALGEBRA. 

EXERaSE  107. 

Determine,  wi43hout  solution,  the  character  of  the  roots  in 
each  equation. 

1.  x'-5x  +  Q  =  0.  8.  2x''  +  Sx  =  5. 

2.  3a:'-7x-2  =  0.  9.3x^-1=2;. 

3.  4x'^4x-  1.  10.  6a;^  -{■^  =  10a;. 

4.  3a:^  +  2a;  +  l=0.  11.  x  =  ^^x' +  1). 

«.  2x'-5x  +  S  =  0.  12.  35a;  +  18  +  12x'  =0. 

6.  9x'  +  12x  +  4=0.  13.  Jx'  =  2a;-3. 

7.  2a;^-f  5a;  +  4=:0.  14.  7a;''  +  l=5a;. 

Determine  the  value  of  m  for  which  the  roots  of  each  equa- 
tion will  be  equal. 

15.  2x'-2x-\-m  =  0.  '              20.  2a;' —  mx  +  12^  =  0. 

16.  2a:'+m  +  a;-0.  21.  ISx" -\- Qx  =  m. 

17.  a:'  +  m  =  3a;.  22.  4a;'^  +  i  =  ma;. 

18.  -ma:'^  — 5a;  +  2=0.  23.  (m  +  l)a;'' +  wa; -- 1. 

19.  5x'  +  Sx-m  =  0.  24.  (?7i  +  l)a;'  +  3m  -  12a?. 

25.  (m  +  l)a;'4-(m-l)a:  +  m  +  l=0. 

26.  2r/uc'^  +  3ma;-7  =  3a;  — 2m  — a;\ 

27.  If  ri  and  r2  represent  the  roots  of  3a;''' —  8a;  +  5  =  0  find  without 
determining  the  actual  roots,  the  values  of: 

ri+r2;  nrg;  rf  H-ri;  n  — r2;  rl  —  rl; 

-8    I   ^3.  JL_1__L_.  _i 1_,     1     r     1. 

7^11-7^2,   n  '    r2»  ri       r2>  rf   '  rl 

28.  Find  the  values  of  the  same  expressions  for  the  equation 
2a;'  — 9ic  +  7  =  0.     Also  for  the  equation  Gx^  —  a;  — 12=0. 

29.  Find  the  values  of  the  same  expressions  for  the  equation 
<ix^  -{-hx-^c  =  0.     Also  for  the  equation  x'  -\-  px  -\-  q  =  0 . 

30.  If  w  and  n  represent  the  roots  of  the  equation  lOa;^  +  9a; — 7=0, 
form  that  equation  whose  roots  shall  be  mn  and  m-j-n.  Form  that 
equation  whose  roots  shall  be  w  —  n  and  -^  -\-  h. 


CHAPTER    XXII. 

RATIO  AND  PROPORTION. 

RATIO. 

273.  The  Ratio  of  two  algebraic  quantities  is  their  exact 
relation  of  magnitude.  Thus,  also,  it  is  the  indicated  quo- 
tient of  the  one  divided  by  the  other,  expressed  either  in  the 
form  of  a  fraction  or  by  the  symbol  :  placed  between  the 
two  quantities.     Thus,  the  ratio  of  a  to  6  is  expressed  as 

^  I. 

- »  or  as  a  :  0. 

0 

274.  The  Terms  of  a  ratio  are  the  two  quantities  compared. 
The  first  term  is  called  the  Antecedent.  The  second  term  is 
called  the  Consequent. 

275.  Bands  of  Ratio.  An  inverse  ratio  is  one  obtained  by 
interchanging  antecedent  and  consequent. 

Thus,  the  direct  ratio  of  a  to  6  is  a  :  6 ;  the  inverse  ratio  of 
the  same  quantities  is  b  :  a. 

A  compound  ratio  is  one  formed  by  taking  the  product  of 
the  corresponding  terms  of  two  given  ratios.  Thus,  ac  :  bd  is 
the  ratio  compounded  of  a  :  6  and  c :  d. 

A  duplicate  ratio  is  formed  by  compounding  a  ratio  with 
itself.     Thus,  the  duplicate  ratio  of  a  :  6  is  a' :  6'. 

In  like  manner  the  triplicate  ratio  of  a  :  6  is  a' :  6'. 

276.  Fundamental  Property  of  Ratios.  If  both  antecedent 
and  consequent  of  a  ratio  be  multiplied  or  divided  by  the  same 
quantity^  the  value  of  the  ratio  is  not  changed. 

^        .  a      ma 

For,  smce  -  =  — -  > 

6       mb 

a :  b  has  the  same  value  as  ma  :  mb, 

291 


296  ALGEBRA. 

PROPORTION. 

277.  A  Proportion  is  an  expression  of  the  equality  of  two 

or  more  equal  ratios. 

^      a      c  ,  , 

JLx.  -  =  -,  or  a\h  =  c'.  d. 
b      d 

278.  Terms  of  a  Proportion.  The  four  quantities  used  in 
a  proportion  are  called  its  terms,  or  proportionals. 

The  first  and  third  terms  are  called  the  antecedents. 

The  second  and  fourth  terms  are  called  the  consequents. 

The  first  and  last  terms  are  called  the  extremes. 

The  second  and  third  terms  are  called  the  means. 

In  a:b  =  c:  d,  d  is  called  a  fourth  proportional  to  a,  6,  and  c. 

279.  A  Continued  Proportion  is  one  in  which  each  con- 
sequent and  the  next  antecedent  are  the  same.     Thus, 

a:b  =  b:c  =  c:d~d:e. 

In  the  simple  continued  proportion  a:  b  =  b:c,b  is  called  a 
mean  proportional  between  a  and  c;  c  is  called  a  third  pro- 
portional to  a  and  b. 

280.  Fundamental  Property  of  Proportion.  For  alge- 
braic purposes  the  fundamental  property  of  a  proportion  con- 
sisting of  four  quantities  is,  that 

The  product  of  the  means  is  equal  to  the  product  of  the  extremes. 

For,  if  a:b  =  c:d, 


XI,         a      c 
then    -  =  -. 

Multiplying  by  bd,          ad  =  be. 

In  like  manner,  if         a:b  =  b:c, 

b'  =  ac.        .'.b  = 

■  Vac. 

This  property  enables  us  to  convert  a  proportion  into  an 
equation,  and  to  solve  a  given  proportion  by  solving  the  equa- 
tion thus  obtained.     (See  Art.  291,  Ex.  1.) 


RATIO  Al^D  PROPORTION.  297 

Before  converting  a  given  proportion. into  an  equation  it  is 
important,  however,  first  to  simplify  the  given  proportion  as 
far  as  possible.  For  this  purpose  we  have  the  following  tran?:* 
formations,  which  are  possible  in  dealing  with  proportions: 

If  four  quantities  are  in  proportion,  they  are  in  proportion  by 

281.  I.  Alternation ;  that  is,  the  first  term  is  to  the  third  as 
the  second  is  to  the  fourth. 

For  if  a:b  =  c:dj 

a_c 
h~d 

Multiplying  by  -»         ~  =  ~.'> 
c  c      d 

.' .  a:c  =  b<d. 

282.  II.  Inversion ;  that  is,  the  second  term  is  to  the  first  as 
the  fourth  is  to  the  third. 

Given,  a:b  =  c:df 

Then  ■    "     " 

Hence, 


b^ 

^~d' 

a 
'b~ 

=  1h 

c 

, 

b 

_d 

a 

c 

b 

:  a  = 

=  d: 

c. 

Or, 

283.    III.  Composition;  that  is,  the  sum  of  the  first  and  second 

terms  is  to  the  second  as  the  su7n  of  the  third  and  fourth  is  to  the  fourth. 

Given,  a:b  =  c:dj 

rw,,  a      c 

Then  -  =  -• 

6      a 

a  c 

Add  1  to  each  side,  -  +  1  =  -  +  1. 
6  a 

.  a  +  b  _  c-]rd 

That  IS,  — ; —  —  — ; —  > 

b  d 

Or,  a-\-b;b=c-\-d;d. 


298  ALGEBRA. 

284.  IV.  Division ;  that  is,  the  difference  of  the  first  and  sec- 
ond terms  is  to  the  second  term  a^  the  difference  of  the  third  and 
fourth  is  to  the  fourth. 


Given, 

a:b=c:d, 

Then 

a      c 

I'd' 

And 

i-H-^ 

That  is, 

a—b       c—d 

b    -  d  ' 

Or, 

a 

—  b:b  =  c  —  d:d. 

285.  V.  Composition  and  Division ;  that  is,  the  sum  of  the 
first  two  terms  is  to  their  difference  as  the  sum  of  the  last  two  terms 
is  to  their  difference. 

Given,                                        a:b  =  c:d, 
Bycomposition(Art.  283),  -^^=-^4^ (1) 


By  division  (Art.  284),         if__[i  _  ii_il ^g) 


6 

'     d       

a-b 

c-d 

b 

~     d       

a  +  b 

c  +  d 

Divide  (1)  by  (2), 

a—o       c—d 

That  is,  a-{-b:a  —  b=c-\-d:c  —  d. 

286.  VI.  Composition  of  Several  Equal  Ratios ;  that  is, 
in  a  series  of  equal  ratios,  the  sum  of  all  the  antecedents  is  to  the 
sum  of  all  the  consequents  as  any  one  antecedent  is  to  its  consequent. 

^.  a      c        e       g 

Let  each  of  the  equal  ratios  equal  r. 

mi.  a  c  e  a 

Then  -=r,    -=r,    -  =  r,    -=r. 

.'.  a  =  br,    c  =  dr,     e=fr,    g  —  hr. 
Adding  the  last  series  of  equalities, 


hatio  and  puoportion.  299 

a  +  c  +  e  -h  ^  =  (6  -r  d  +  /  +  A)r. 
•     Q  +  c  H-  e  +  fl^         _a 
'  b  +  d-\-f+h  ~^~b' 
.'.  a-{-c-{-e  +  g:b  +  d+f+h-=a:b. 

287.    VII.  Product  of  Corresponding  Terms.     In  two  or 

more  proportions  the  products  of  the  corresponding  terms  are  in 
proportion. 

Given,  a:b  =  c:dy 

j:k  =  l:m. 


Then 

a  ^c       e       g      j      I 
b      d      f      h      k      m 

Taking 

the 

product  of  corresponding  members  of  these 

equations, 

oe;         cgl 

bfk       dhm 
,' .  aej :  bfk  =  cgl :  dhm. 

288.    VIII. 

Powers  and  Roots.     In  any  proportion  like 

powers  or 

like 

roots  of  the  terms  are  in  proportion. 

Given, 

a:b  =c:d. 

Then 

a  _c 
b~d 

I         I 

Hence, 

^  =  ^-        Also,-=-., 
b-      d- 

That  is, 

^n.jn^^n.^n^ 

And 

11            11 

a"" :  b""  =  c"" :  d\ 

289.    IX.  Cancellation  of  Factors  of  Terms.    From  Arts. 
276  and  281  it  is  evident  that  if  four  quantities  be  in  proportion, 


300  ALGEBRA. 

and  if  the  first  two  terms  or  the  last  two^  or  the  first  and  third,  or 
second  and  fourth,  be  multiplied  or  divided  by  the  same  quantity, 
the  resulting  quantities  are  in  proportion. 

Thus,  if  a'.b  =  c:d, 

Then  ma  :mb  —  nc:  nd. 

And  ma  :  pb  =mc:  pd. 

290.  Equal  Products  made  into  a  Proportion ;  that  is,  if 

the  product  of  two  quantities  is  equal  to  the  product  of  two  other 
quantities,  either  two  may  be  made  the  means,  and  the  other  two  the 
extremes  of  a  proportion. 

For,  if  ad  =  be, 

Dividing  hy  bd,  -  =  -. 

b      d 

.  * .  a:b  =  c:  d. 

This  property  is  evidently  the  converse  of  the  principle 
stated  in  Art.  280. 

291.  Application  of  these  Principles.  The  use  of  propor- 
tion in  solving  algebraic  problems  and  determining  the  prop- 
erties of  algebraic  quantities  may  be  reduced  essentially  to 
the  following: 

T.  By  taking  the  product  of  the  means  equal  to  the  product  oj  the 
extremes,  a  proportion  may  be  converted  into  an  equation,  and  the 
proportion  solved  by  solving  the  equation. 

Ex.  1.   Find  the  value  of  x  which  satisfies  the  proportion, 

4a;  -  1  :  a;  -I-  1  =  3x  +  1 :  2a;  —  1. 

Taking  the  product  of  the  means  equal  to  the  product  of 
the  extremes, 

(4a;- 1)  (2x- 1)  -  (^  +  1)  (3x  +  1) 
.-.  5a;^-10a;  =  0 

x  =  0,  2. 


RATIO  AND  PROPORTION.  301 

292.  II.  Before  converting  a  'proportion  into  an  equation  it  is 
important  to  simplify  the  proportion,  as  far  as  possible,  by  use  of  the 
properties  of  a  proportion,  as  Alternation,  Composition,  Division,  etc. 

Ex.1.  Solve  x'  —  2a;  +  3:a;'  +  2a;-3  =  2x'-a;-3:2x'  +  a;  +  3. 

By  Composition  and  Division,  ^^ 


Divide  by  1x\ 


4a;  -  6      2a;  +  6 
1  2 


2a;  -  3      a;  +  3 
.-.  a;  +  3  =  4a;-6 
a;-3 
The  factor  2a;'^  divided  out  also  gives  the  roots  a;  =  0,  0. 

VxTl  +  Vx^^^l      4x-l 


Ex.  2.   Solve 


VxTl  -  Vx"=^l  2 


By  Composition  and  Division,      ^  "^  4a;  +  I 


4a; -3 

16a;'  +  8a;  +  1 
1  16a;2-24a;  +  9 
X      16a;2  -  8a;  +  5 


Squaring, 

By  Composition  and  Division: 

1  16a;  —  4 

Hence,  16x'  -  4a;  =  Ux^  -  8a;  +  5 

a;  =  |. 

293.     III.  Given  some  proportion  (or  equality  of  several  equal 

ratios),  as  a  :  b  =  c :  d,  a  required  proportion  is  often  readily 

ft      f» 
proved  by  taking  -  =  -  ~x  (hence,  a  =  hx,  c  —  d-x),  and  sub- 

b      d 

stituting  for  a  and  c  in  the  required  proportion. 

Ex.  Given,  a:b  =  c:  d, 

Prove  2a'  +  Bab' :  2a'  -  Sab'  =  2(f  +  3cd' :  2c'  -  3ccP. 

Let  ^  =  £  =  x,        .-.  a  =  bx,  G^  dx. 

o      a 

Substitute  in  each  ratio  the  values  a=  bx,  c=  dx. 

2a^  +  3aP      2h^a^'+  3b^x       bh'{2x^  +  3)       2a;'  +  3 


I. 
II. 


2a^  -  3rt6'  ~  2b^x^  -  Sb^'x       6"^a;(2ar*  -  3)       2x''  -  3' 
2c^  +  Scd^  ^  2d'x^  +  Wx  ^  d'^xj^x^  +  3)  _^  2x^  ^  3 
2c*  -  'Scd'  "  la'x'  -  3d'x      d^x(2x'  -  3)     2a!'  -  3 
2d^  4-  Sab""      2c^  +  Scd^ 


'  ••  2a^  -  3ab^      2c'  -  3cd?' 
since  they  are  each  ecjual  to  the  same  expression. 


a02  ALGEBRA. 


EXERCISE  108. 

Find  the  ratio  of  x  to  y — 

1.  lx-2>y  =  ^x  +  y.                   ^    Sx-2y 

o.                        — 

2.  4:X  —  5y:5x  —  Ay—  f .                  4x  —  Sy 

a 

4.  x^  +  62/^  =  5xy. 

Find  a  mean  proportional  between — 

6.  Sab'  and  12a'.      6.  3^^  and  2|.      7.  (a  —  x)'  and  (a  +  aj)*. 


3x'-5a;-12        ,        3x' 4- 4a; 

8. and 

^x'  +  5x                Sx''-4x--15' 

21/6  +  51/3 

9.  ■ and 

31/2-4 

31/6-41/3 

81/2  +  20 

Find  a 

fourth  proportional  to— 

10. 

2a,  36,  4ac. 

12.  h  f ,  A. 

11. 

x\  xy,  ^x'. 

13.  a  — 1,  a,  1. 

Find  a 

third  proportional  to — 

14. 

x  and  5. 

16.  (a  +  iy  and  a'- 

"1 

15. 

1^  and  7^. 

1       ^  1      . 
17.  a--  and  --1. 

a          a 

Solve  the  equations — 

18.  2a:  +  3  :  3x  -  1  =  3x  +  1  :  2a;  +  1. 

19.  a;  +  5:3-a;  =  10  +  3x:a;-10. 

20.  3a;  +  5:5x  +  ll=7— x-.-Sx. 

21.  a;'-4:a;'-a;  +  3  =  a;  +  2:2x  +  3. 

22.  x'  +  2x-l:2:'  +  2a;  +  5=:2x+l:2a;--5. 

23.  a;'-3x'  +  5:x'  +  3x'-5  =  a;'  +  2:a;'-2. 

24.  2a;'  -  8a;'  -  3x  +  1 :  2a;'  -  10a;'  +  3a;  -  1  =a;'  + 11 :  x'-ll. 

25.  VWxl^:2VW=^==l/x=l:VxTT. 
3  +  V2xT'^      4  +  VxTT 


6  -  i/^x+'B      4 -  l/x+1 


RATIO  AND  PROPORTION.  303 


^    3a  +  VAx - 3a'       a  +  VxT^ 
27. 


6a  —  l/4x^ 3a^       3a  -  VxT^ 
28.  82/-6x:a;  +  2/-l=5-3x:4-2/  =  7;4. 

Ui/  — 3:«  — l=a  +  2:l. 
If  a  :  6  =  c :  d,  prove— 

30.  a^-.c'^ab'.dc.  31.  a' :  6'-a*  +  c':  6»4- (P. 

32.  ac:6d  =  (a  +  cy:(6  +  d)l 

33.  {a-  cy  :(h  --ay  =  a^  +  c'  '.h'  -{■  d^. 

34.  a  :  6  =  VaTTZ^  :  V¥~+W, 

35.  2a'  +  3a6  :  3a6  -  46'  =  2c'  +  Scd :  Zed  -  4(f . 

36.  a'-a6  +  6':^^^ ^  =c'-cci  + d': -^ -- 

a  e 

If  a,  b,  c,  d  are  in  continued  proportion,  prove — 

37.  a:c-d  =  b':bd-cd. 

38.  a :  c  =  a'  +  6'  +  c' :  6'  -f  c'  +  d\ 

39.  a  :  d  =  a'  +  26»  +  3c' :  b'  +  2c'  +  3cf . 

Prove  that  a  :  6  =  c :  c?,  it  being  given  that — 

40.  (a  +  b)  (c  —  d)  +  (6  +  c)  (d  —  a)  =  C(i  -  a6. 

41.  (a  +  6-3c-3d)(2a-26-c  +  (i)  =  (2a  +  26-<;-(0 
(a-6-3c  +  3c?). 

42.  Find  two  numb^s  in  the  ratio  of  2  to  5,  such  that 
when  each  is  increased  by  5  they  shall  be  as  3  to  5. 

43.  Find  two  numbers,  such  that  if  7  be  added  to  each 
they  will  be  in  the  ratio  of  2  to  3 ;  and  if  2  be  subtracted 
from  each,  they  will  be  in  the  ratio  of  1  to  3. 

44.  Separate  32  into  two  parts,  such  that  the  greater  dimin- 
ished by  11  shall  be  to  the  less,  increased  by  5,  as  4  to  9. 

45.  Separate  12  into  two  parts,  such  that  their  product  shall 
be  to  the  sum  of  their  squares  as  2  to  5. 


CHAPTER    XXIII. 

INDETERMINATE  EQUATIONS.    VARIATION. 

294.  Indeterminate  Equations.  If  a  single  equation  con- 
taining two  unknown  quantities  be  given,  this  equation  is 
called  an  indeterminate  equation^  for  the  unknown  quantities 
may  have  an  indefinite  number  of  different  values  which 
satisfy  the  ^quation. 


Thus,  given 

3a;  +  22/  =  5. 

When 

r  =  0, 

2/  =  f, 

x  =  l, 

2/  =  l,. 

x  =  2, 

2/=-i 

x  =  Z, 

2/=-2, 
etc. 

In  an  indeterminate  equation  some  limitation  in  the  char- 
acter of  the  values  of  x  and  y  may  be  imposed.  Very  fre- 
quently the  values  of  x  and  y  are  limited  to  positive  integers. 
Of  the  values  obtained  for  x  and  y  in  the  above  equation,  the 
only  set  that  satisfies  this  condition  is  x  =  1,  y  —  1. 

In  like  manner,  if  in  a  group  of  given  simultaneous  equa- 
tions the  number  of  unknown  quantities  be  greater  than  the 
number  of  the  equations,  the  equations  are  said  to  be  inde- 
terminate. 

The  treatment  of  the  subject  here  made  wdll  be  limited  to 
indeterminate  equations  of  the  first  degree. 

295.  The  Solution  of  Indeterminate  Equations  is  best 
explained  in  connection  with  illustrative  examples. 

304  '  '  ■ 


INDETERMINATE  EQUATIONS,  306 

Ex.  1.   Solve  in  positive  integers  5z  —  7y  =  11. 
Divide  through  by  5,  the  smaller  of  the  two  coefficients. 

5 

5 

Since  x  and  y  are  integers,  x-  y  -2  must  be  an  integer.     Hence, 

-^ must  be  an  integer. 

5 

.    3(2.y  +  1)  _  6w  +  3         , , 
•    •  ~^ — -^*  or  -»2— — ,  must  be  an  mteger. 
o  o 

(The  particular  multiplier  3  is  used  in  this  case  so  that  on  dividing  the 
resulting  numerator  by  the  denominator  5,  the  coefficient  of  y  in  the  re- 
mainder is  unity.) 

-^- —  ;    hence,  y  +  ^^—  '    hence,  2L±_?  must  be  an  integer, 
o  o  5 

Let  ^^^^=p. 

•      .-.^  =  5^-3 (1) 

Substitute  in  the  original  equation  for  y, 

5x  -  35p  +  21  =  11 

.'.  x  =  7p-2 (2) 

In  equations  (1)  and  (2)  p  must  have  some  integral  value. 
If    p  =  1,         then    x  =  5,       y  =  2. 
If    p  =  2,  a;  =  12,      2/  =  7. 

Etc.  etc. 

It  is  seen  that  there  are  an  indefinite  number  of  positive  integral  values 
of  X  and  ?/. 

Ex.  2.  A  number  consists  of  two  digits ;  if  the  number  be 
divided  by  the  number  formed  by  reversing  the  digits,  the 
quotient  is  2,  and  the  remainder  2.     Find  the  number. 

Let  X  =  the  tens'  digit  ^ 

y  =  the  units'  digit 

Then  ^Q^  -^  y  -  ^  =  2 

x  +  IQy 

29 


306  ALGEBRA, 

.  • .  8a:  -  19?/  -  2. 
Dividing  by  8,      a;  -  2^/  -  ^  -  | 

....-2^  =  §^ 

.  • .  -^ must  be  an  integer. 

Hence,  -^-^ ^  >    or  -^ >  and  ^ must  be  int-egera. 

'8  8  8  ^ 

Let    y-^=p 
8  ^ 

\x=  19p-  14 

The  values  of  a:  as  digits  in  a  number  are  limited  to  positive  integers, 
lowest  0,  highest  9. 

.  • .  a:  =  5,  2/  =  2  is  the  only  result  allowable, 
.  • .  the  number  is  52. 

Ex.  3.  In  how  many  ways  can  the  sum  of  $5.10  be  paid 
with  half-dollars,  quarters,  and  dimes,  the  whole  number  of 
coins  used  being  20? 

Let  X  =  number  of  half-dollars. 

y  =  number  of  quarter-dollars. 
z  =  number  of  dimes. 

'^'^  f^f  +  fo  =  **- 

Or,  lOx  +  5^  +  2^  =  102 (1) 

Also,  a;  -h  ?/  +  0  =  20 (2) 

Multiply  (2)  by  2,  and  subtract  from  (1), 
8a:  -h  3?/  =  62 

r  a:  =  3p  +  1. 


Solving, 

■  y  = 

18 
5p 

-  %p. 
+  1. 

Let 

V- 

-0, 

then 

x=l, 

y 

=  18, 

z  =  \. 

P- 

=  1, 

a:  =  4, 

y 

=  10, 

z  =  6. 

P  = 

=  2, 

a:  =  7, 

y  ■ 

=  2, 

z  =  11. 

Any  other  values  of 

p  give 

negative 

results  for  one  or  more 

of  th« 

quantities  a:,  ?/,  z. 
Hence,  tbere  are 

thret 

)  ways  of 

'  making 

the 

required  payment. 

VABIATION.  307 

EXERCISE  109. 

Solve  in  positive  integers — 

1.  7x  4-  42/  -  63.  6.  lOx  +  17y  =  199. 

2.  3x  +  lly  =  31.  7.  5x~7y  =  11. 

3.  5x  +  72/  =  82.  8.  132;  -  SOy  =  61. 

4.  7a; +  122/ -111.  9.  16a;  -  Ht/ =  26. 

5.  15a;  +  82/ -  101.  10.  13a;  -  352/ -  -  64. 

11.  Divide  the  number  107  into  two  such  parts  that  one  is 
divisible  by  3,  and  the  other  by  8. 

12.  Divide  321  into  two  such  parts  that  one  is  divisible  by 
9,  and  the  other  by  13. 

18.  Find  two  fractions  whose  denominators  are  5  and  12 
respectively,  and  whose  sum  is  4^V 

14.  A  farmer  sold  a  number  of  sheep  and  calves  for  $194 ; 
for  each  sheep  he  received  $6,  and  for  each  calf  $11.  How 
many  of  each  did  he  sell? 

15.  In  how  many  ways  can  the  sum  of  $5.80  be  paid  with 
dimes  and  quarters? 

16.  Find  all  possible  ways  of  paying  three  dollars  with 
five-,  ten-,  and  twenty-five-cent  pieces,  so  that  half  the  coins 
used  are  five-cent  pieces. 

17.  There  is  a  number  which,  when  divided  by  17  gives  a 
remainder  of  6,  and  when  divided  by  23  gives  a  remainder 
of  21.    Find  it.     How  many  such  numbers  are  there? 

VARIATION. 

296.  Variables  and  Constants.  A  Variable  is  a  quantity 
which  has  an  indefinite  number  of  different  values. 

A  Constant  is  a  quantity  which  has  a  single  fixed  value. 

297.  Relation  of  Variables.  Variations.  One  variable 
(called  the  function)  may  depend  on  another  variable  for  its 
value  in  a  definite  manner.    Thus,  if  a  man  be  hired  to  work 


308  ALGEBRA. 

for  a  certain  sum  per  day,  the  number  of  dollars  he  will  re- 
ceive as  wages  will  vary  as  the  number  of  days  he  works. 

Thus,  if  a;  =  number  of  dollars  in  his  wages, 
t  —  number  of  days  he  works, 
X  oc  t.     (The  symbol  a  reads  "  varies  as.") 

This  expression  is  called  a  variation. 
This  variation  may  also  be  expressed  thus, 

X  =  mty 

where  m  denotes  the  number  of  dollars  in  one  day's  wages. 

Or,  -  =  m. 

t 

Thus,  if  the  ratio  of  two  variables  is  always  constant,  their 
relation  may  be  expressed  in  any  one  of  three  ways : 

(1)  As  a  ratio. 

(2)  As  an  equation. 

(3)  As  a  variation. 

KINDS  OF  ELEMENTARY  VARIATIONS. 

298.     I.  Simple  Direct  Variations.     The  case  considered 
m  Art.  297, 

X  <x  y,     OT  x  =  my  J 

is  called  a  direct  variation. 

II.  Inverse  Variations.     If  x  varies  inversely  as  y  (that  is, 
as  X  increases  y  decreases,  and  vice  versd)^  then  x  and  -  have 

y 

a  constant  ratio, 

1  m 

X  cc  -1    or  x  =  —  • 

y  y 

This  is  called  an  inverse  variation. 
Thus,  the  number  of  days  required  in  which  to  do  a  given 


VARIATION.  309 

piece  of  work  varies  inversely  as  the  number  of  workmen 
employed.  Also,  in  triangles  of  a  given  area  the  altitude 
varies  inversely  as  the  length  of  the  base. 

III.  Joint  Variation.  If  x  varies  as  the  product  of  two  or 
more  other  variables,  as  of  y  and  z,  for  instance,  then  x  and  yz 
have  a  constant  ratio,  and 

X  (X  yz,     or  a;  =  myz. 
This  is  called  a  joint  variation. 

IV.  Direct  and  Inverse  Variation,  x  may  also  vary  di- 
rectly as  one  variable,  as  2/,  and  inversely  as  another,  as  z ; 
then 

X  cc  -y     or  a;  = 

z  z 

299.  Compound  Variations.  The  sum  or  difference  of 
two  or  more  variations  may  be  taken,  the  result  being  termed 
a  compound  variation. 

Thus,  if  y  equals  the  sum  of  u  and  v,  and  u  varies  directly 
as  x'j  and  v,  inversely  as  x, 

u  =  mx .    V  =  -  > 
'  X 

y  =  u  +  v. 

n 

.' .  y  =  mx  +  -  • 
^  X 

800.  Fundamental  Property  of  Variations.  A  variation 
may  be  converted  into  an  equation  by  the  use  of  a  coefficient  which 
is  afterward  to  be  determined,  and  the  properties  of  variations  de- 
rived and  problems  solved  by  the  use  of  the  properties  of  equations. 

301.  Elementary  Properties  of  Variations. 
I.  If  X  a  2/,     and  y  cc  z,    then  x  cc  z. 
For    x  —  my,        y—rvz, 
,' .  x  =  mnz. 
,' .  X  ocz. 


310  ALGEBRA. 

II.  If  a;  a  z,     y  oc  z,     then  x±:y  ca  z    and  Vxy  tjf^  25, 

For  X  =  mz,        y  =  nz. 

.  * .  a;  dz  2/  =  (m  zt  n)z, 
And  V^  =  l/mz •  nz  =  Vrm^  =  z l/mn. 
Hence,  x±y  oz  z. 

Vxy  oc  z. 

III.  If  a;  a  z,    and  y  oc  u,    then  xi/  a  wz. 

For  x  =  mz.        y  =  nu. 

.  * .  iC2/  =  mnuz. 
.  • .  a;2/  oc  i(z. 

IV.  If  a;  a  2/,     then  a;"  oc  3/". 

For  X  =  ray. 

.  * .  a;*'  =  m'*2/'*. 
.  • .  a;"  oc  2/**. 

302.  Examples. 

Ex.  1.   If  a;  varies  inversely  as  2/',  and  a;  =  4,  when  y  =»  I, 


find  a;  when 

y-- 

=  2. 

Since 

xcc  —, 

y' 

we  have 

y^ 

Substitute  x  = 

=  4, 

2/  = 

1,  in 

(1): 

,  4  =  m. 

Substitute  for 

m 

its  value 

in 

(1), 

-^ 

(1) 


(2) 

Let  2/  =  2  in  (2),     then  a;  =  1,  Result. 
Ex.  2.   If  2/  equals  the  sum  of  two  quantities,  one  of  which 


VARIATION,  311 

varies  directly  as  a;',  the  other  inversely  as  x;  and  y  =  5  when 
a;  =  1, 2/  =  1  when  x  =  —  1,  find  y  when  ic  =  2. 


Since 

2/  =  M  +  V,     and  u  oc  a;^, 

VOC  —' 
X 

[Art.  299.] 

X 

(1) 

Substituting  the  given  pairs  of  values  for  x  and  y  in  (1), 

5  =  m  +  n. 

1  =  m  —  n. 
.  • .  m  =  3,        n  =  2. 
Substitute  in  (1)  for  m  and  n, 

2/  =  Sx^'  +  I (2) 

Let  a;  =  2  in  (2),  ^  =  13,  Result. 

Ex.  3.  The  area  of  a  circle  varies  as  the  square  of  its  diam- 
eter. Find  the  diameter  of  a  circle  whose  area  shall  be  equiv- 
alent to  the  sum  of  the  areas  of  two  circles  whose  diameters 
are  6  and  8  inches  respectively. 

Let  A  denote  the  area  of  a  circle,  and  D  the  diameter. 
Then  A  ccB'', 

And  A  =  ml)\ 

denote  the  areas  of  the  two  given  circles  by  ^''  and  A'^» 
Then  ^/  =  ryi  x  6^  =  36m. 

^//  =  771  X  8^  =  6im, 
Adding,  A^  -\-  A^^,  or  A  =  100m, 
Hence,  since  A  =  inD"^,  and  also  100m, 
mD"^  =  100m 
i)2  =  100 
D  =  10 
Thus,  the  required  diameter  is  10. 

The  student  should  review  examples  1,  2,  and  3  thoroughly,  until  he 
understands  every  step  taken  in  their  solution,  before  he  undertakes  a 
single  example  of  the  following  exercise. 


312  ALGEBRA. 

EXERCISE  110. 

1.  If  X  varies  as  3/,  and  x  is  10  when  y  is  2,  find  x  when  y 
is  3. 

2.  If  a:  ex  3/,  and  a:  =  8  when  2/  =  6,  find  y  when  x  =  3. 

3.  If  a;  + 1  a  2/  —  5,  and  a;  =^  2  when  2/  =  6,  find  a;  when  2/  =  7. 

4.  If  x^  oc  2/',  and  a;  =^  4  when  2/  =  2,  find  y  when  a;  =  32. 

5.  If  x^  a  2/'^  -f  8,  and  a;  =  f  l/S  when  2/  =  1?  fi^^d  2/  when 
a;  =  3. 

6.  If  X  varies  inversely  as  2/,  and  equals  2  when  y  is  4,  find 
2/  when  a:  =  5. 

7.  If  a;  varies  inversely  as  2/^  and  is  6  when  y  is  |,  find  y 
when  a;  =  lj. 

8.  If  X  varies  jointly  as  y  and  z,  and  is  6  when  2/  is  3  and  2 
is  2,  find  a;  when  2/  is  5  and  z,  7. 

9.  If  X  varies  jointly  as  2/  and  z,  and  equals  2  when  y  =  \ 
and  2=1,  find  x  when  2/  =  3,  z  =  f. 

10.  a:  varies  directly  as  y  and  inversely  as  2,  and  =  10  when 
2/  =  15  and  2  =  6.     Find  y  when  a;  =  16  and  2  =  2. 

11.  If  2a:  —  32/  a  5a;  +  92/,  and  when  2/  =  —  2,  a;  =  4,  find  the 
equation  connecting  x  and  2/. 

12.  If  a;'-2a;  +  l  cxy''-2y-\,  and  a;  =  |  when  y=-h 
find  the  equation  between  x  and  2/- 

13.  One  quantity  varies  directly  as  x  and  another  varies 
inversely  as  x.  If  their  sum  is  equal  to  10  when  a;  =  2,  and 
to  —  2  when  a;  =  — 1,  find  each  quantity  when  a:  =  f. 

14.  Two  quantities  vary  directly  as  x^  and  inversely  as  x 
respectively.  If  their  sum  is  3^  when  a;  =  2,  and  — 3J  when 
a;  =  1,  find  the  quantities  in  terms  of  x. 

15.  Given  that  w  is  equal  to  the  sum  of  two  quantities 
which  vary  as  x  and  a;',  respectively.  If  w  =  —  2  when  a;  =  2, 
and  —5  when  a;=  — 1,  what  is  w  when  x  =  |? 

16.  Given  that  w  is  equal  to  the  sum  of  two  quantities 
which  vary  as  x  and  x^  respectively.  liw=~2  when  a;  =  —  1, 
and  w  =  —  ll^  when  a;  =  —  2,  what  is  the  value  of  w  when 


VARIATION.  313 

17.  Given  that  w  is  equal  to  the  sum  of  three  quantities, 
one  of  which  is  constant  and  the  others  vary  directly  as  x* 
and  inversely  as  x^  respectively.  If  w  =  S  when  x  =  ^jW  =  S 
when  x=  —  Ij  and  w  =  16f  when  x  =  —  |,  find  the  equation 
between  w  and  x. 

18.  The  distance  fallen  by  a  body  from  a  position  of  rest 
varies  as  the  square  of  the  time  during  which  it  falls.  If  a 
body  falls  144|  feet  in  3  seconds,  how  far  will  it  fall  in  8 
seconds  ? 

19.  The  area  of  a  circle  varies  as  the  square  of  its  diam- 
eter. Find  the  diameter  of  a  circle  equivalent  to  two  circles 
whose  diameters  are  5  and  12  inches  respectively. 

20.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diam- 
eter. If  three  spheres  whose  diameters  are  respectively  6,  8, 
and  10  inches  be  formed  into  a  single  sphere,  find  its  di- 
ameter. 

21.  The  volume  of  a  cone  of  revolution  whose  altitude  is  7, 
and  the  radius  of  whose  base  is  3,  is  66.  Find  the  volume 
of  a  cone  of  revolution  of  altitude  6  and  radius  5. 

Note.  The  volume  of  a  cone  of  revolution  (or  cylinder  of  revolution) 
varies  jointly  as  the  altitude  and  the  square  of  the  radius  of  the  base. 

22.  Find  the  altitude  of  a  cone  of  revolution  the  radius  of 
whose  base  is  7,  and  which  is  equivalent  to  two  cones  with 
altitudes  5  and  11  and  radii  2  and  4  respectively. 

23.  If  the  illumination  from  a  source  of  light  varies  in- 
versely as  the  square  of  the  distance,  how  much  farther  from 
a  candle  must  a  book  which  is  now  18  inches  away,  be 
removed  to  receive  just  J  as  much  light?  Interpret  the 
two  results. 


CHAPTER    XXIV. 
ARITHMETICAL  PROGRESSION. 

303.  A  Series  is  a  succession  of  terms  formed  according  to 
some  law. 

Exs.   1,  4,  9,  16,  25, 

l-x  +  x'-^-i-x'-, 

2,  4,  8,  16,  32, 

304.  An  Arithmetical  Progression  is  a  series  each  term 
of  which  is  formed  by  adding  a  constant  quantity,  called  the 
difference,  to  the  preceding  term. 

Thus,  1,  4,  7,  10,  13, is  an  arithmetical  progression 

in  which  the  difference  is  3. 

Given  an  Arithmetical  Progression  (often  denoted  by  A.  P.), 
to  determine  the  difference,  from  any  term  subtract  the  preceding 
term. 

Thus,  in  the  A.  P.,        |,  -  |,  -  3, 
the  difference  =  —  |— f  =  —  |. 

305.  Principal  Quantities  and  Symbols  Used.  In  an 
A.  P.  we  are  concerned  with  five  quantities : 

1.  The  ^rs^  term,  denoted  by  a. 

2.  The  common  difference,  denoted  by  d. 

3.  The  last  term,  denoted  by  I.  "" 

4.  The  number  of  terms,  denoted  by  n. 

5.  The  sum  of  the  terms,  denoted  by  s. 

306.  Two  Fundamental  Formulas.  Since  in  an  A.  P. 
each  term  is  formed  by  adding  the  common  difference,  d,  to 
the  preceding  term,  the  general  form  of  an  A.  P.  is — 

a,     a  +  d,     a  +  2(i,     a  -f  3(i,  -f 

314 


ARITHMETICAL  PROGRESSION.  315 

Hence,  the  coefficient  of  d  in  each  term  is  one  less  than  the 
number  of  the  term. 

Thus,  the  7th  term  is  a  +  6c?, 

12th  term  is  a  +  lie?, 
nth  term  is  a  +  (n  —  l)d. 

Hence,  l  =  a-i-(n-l)d (1) 

Also, 

s  =  a  +  (a  +  d)  +  (a-^2d)+ -\-(l-d)  +  l  .    .(2) 

Writing  the  terms  of  this  series  in  reverse  order, 

s='-l  +  (l-d)  +  il-2d)+ ^+(a  +  d)  +  a    .    .(3) 

Adding  (2)  and  (3), 

2s  =  (a  +  0  +  (a  +  0  +  (a  +  0  + +  (a  +  0  H-  (a  +  0 

=  n(a  +  0. 

.•.s  =  |(a  +  ?) (4) 

If  we  substitute  for  I  in  (4)  from  (1), 

s-^[2a  +  (n-l)c?] (5) 

Hence,  combining  results,  we  have  the  two  fundamental 
formulas  for  I  and  s, 

I.  l  =  a  +  (n-l)d, 
II.  s  =  ^(a  +  l) 

s=-[2a  +  (n-l)fq. 

Ex.  1.   Find  the  12th  term  and  the  sum  of  12  terms  of  the 

A.P,  5,3,1,-1,-3, 

In  this  series  a  =  5,     d  =  -  2,     n  =  12. 

From  I.,  i  =  5  +  (12  -  1)  (-  2)  =  5  -  22  =  -  17. 

From  II.,         8  =  -»j«(5  -  17)  =  -  72,  Sum. 


316  ALGEBRA. 


Ex.  2.   Find  the  sum  of  n  terms  of  the  A.  P., 

a  +  b       a  —  b       a  —  36 
■— — : —  > 


2  2 

Here  a  = >    d  =  —  b,     n  =  n. 

Substituting  in  the  fundamental  formula,  s  =  -  [2a  +  (n  —  l)(i], 

A 

«=|[a  +  6  +  (n-l)(-6)] 
=  ^[a  +  (2  -  n)6],  ^um. 

EXERCISE  111. 

1.  Find  the  8th  term  in  the  series  3,  7,  11, 

2.  Find  the  9th  term  and  the  sum  of  9  terms  in  7,  3,  —  1, 

3.  Find  the  13th  term  and  the  sum  of  13  terms  in  — 10, 
-13,-16, 

4.  Find  the  20th  and  28th  terms  in  5,  -^,  ^^,  .  .  .  .  . 

5.  Find  the  16th  and  25th  terms  in  —  13|,  -  9,  -  4| 

6.  Find  the  7th  and  10th  terms  and  the  sum  of  10  terms  in 
the  series  |,  |,  A, 

7.  Find  the  18th  term  and  the  sum  of  18  terms  in  the 
series  3,  2.4,  1.8, 

Find  the  sum  of  the  series — 

8.  3,  8,  13, ....  to  8  terms. 

9.  —  4,  —  7,  — 10, .  .  .  .  to  6  terms. 

10.  3,  —  3,  -  9, ....  to  9  terms. 

11.  21,  3f ,  5, .  .  .  .  to  14  terms. 
12-  |j  i?  I,  •  •  •  •  to  96  terms. 

13.  —  i,  i,  f, .  .  .  .  to  38  terms. 

14.  —  f ,  —  f,  —  lij  •  •  •  •  to  55  terms. 

15.  3c,  1^,  —  2c, ....  to  6  terms. 

16.  2a:  —  3/,  a;  +  2/,  3^/, .  .  .  .  to  r  terms. 

17.  5l/2-2v^,  4V/2-31/3,   ....  to  11  terms. 

18.  3a )  2a,  a  H-  -, .  .  .  .to  12  terms. 


ARITHMETICAL  PROGRESSION.  317 

307.  Problem  I.  Given  any  three  oj  the  five  qiiantities^  a,  d, 
If  n,  s,  to  find  the  other  two. 

If  we  substitute  for  tbe  three  given  quantities  their  values 
in  the  two  fundamental  formulas  (I.  and  II.,  Art.  306),  we 
shall  have  as  a  result  two  equations  with  two  unknown  quan- 
tities. The  values  of  these  unknown  quantities  may  then  be 
found  by  solving  the  two  equations.  Hence,  by  the  use  of 
these  fundamental  formulas,  problems  relating  to  A.  P.  are 
converted  into  problems  relating  to  the  solution  of  equations, 
processes  already  mastered. 

Ex.   Given  d  =  2,  1=21,  s  =  121,  find  a,  n. 
Substitute  for  d,  I,  sin  Formulas  I.  and  II., 

21  =  a+  (n-l)2 .(1) 

121  =  ^(«±^ (2) 

2 

.  • .  a  +  2n  =  23 (3) 

ari  +  21n  =  242 (4) 

Substitute  for  a  in  (4)  from  (3), 

«(23  -  2n)  +  21n  =  242 
Whence  n  =  11 

Hence,  from  (3),  a  =  1. 

308.  Problem  11.  Given  three  of  the  five  quantities,  a,  d,  I, 
n,  8,  to  obtain  a  formula  for  one  or  both  of  the  other  two  in  terms 

of  the  three  given  quantities. 

Ex.  Given  d,  I,  s,  obtain  a  formula  for  n. 

Since  I  =  a -^  [n  -  \)d (1) 

s  ==  ^[2a  +  (n  -  l)d] (2) 

Substitute  for  a  from  (1)  in  (2), 

8  =  ^[2^  -  2(n  -  l)d  +  (n  -  l)d] 

.' .  2s  =  2ln-  n{n  -  \)d 

Whence  dn>  -  (d  +  2l)n  =  -  2s 

c  ,  .      .                                 d  +  2^  ±  V{d  +  2/)'  -^"8^. 
Solvmg  for  n,  n= —z '■ • 


318 


ALGEBRA. 


No. 


Given. 


Requiked. 


Formulas. 


a,  d,  n 
a,  d,  s 


a  +  {n-  l)d. 

_  irf  i  i/2ds  +  {a-  idy. 

a. 

n 

s       {n  —  V)d 
n  2 


n,  I,  s 
d,  n,  s 


I-  {n-  \)d. 

\d  ±  V{\d  +  If  -  2ds. 

n 

s  _  (n  —  Vjd  ^ 
n  2 


10 

11 
12 


a,  d,  n 
a,  d,  I 

a,  n,  I 
d,  n,  I 


=  ln[2a  +  (n  -  l)d]. 
=  ^+  «  +  ^^  -  aV 


2d 


^{a  +  l). 


8 

s  =  \n\2l  -  (n  -  l)d]. 


13 
14 
15 
16 


a,  I,  s 
n,  I,  s 


I  —  a^ 
n-i 

2(g  -  an) 
n[n  —  1) 


2s-  I-  a 

2{nl  -  s) . 
n(ri  —  1) 


17 
18 
19 
20 


Of,  c?,  ? 
a,  d,  s 
a,  ^,  s 
d,  ^,  s 


I  —  a 


+  1. 


_  rf  -  2a  Jk  l/(d  -  2af  +  Sdg 


2d 


2s 


I  +  a 

2^+  d=fcl/(2^  +  d)'-'-  8d.9. 
2d 


ARITHMETICAL  PROGRESSION.  319 

EXERCISE  112. 

Find  the  first  term  and  the  sum  of  the  series  when— 
1.  d  =  3,  1  =  40,  71  =  13.  2.  d  =  l  1  =  1SI  n  =  ZB. 

Find  the  first  term  and  the  common  difierence  when— 
3.  s  =  275,  ^  =  45,  n  =  ll.         4.  5  =  4,  /=-10,  n  =  8. 
6.  s=-246i,  ^=-34^,  71  =  17. 

6.  s=-38i,  l=-H,  ri  =  21. 

7.  s  =  9,  /  =  2|,  71  =  9. 

8.  5=-^,  Z=-4,  71  =  47. 

Find  71  and  d  when — 
9.  a  =  -5,Z  =  15,s  =  105.  11.  a=^,l= -^,  $  =  -2^. 

10.  a  =  19,^  =  -21,s  =  -21.        12.  a= -3J,  ^  =  9J,  5=48, 

Find  a  and  7i  when — 

13.  ;=-8,d  =  -3,5  =  -3.        15.  l  =  2,d=-i,s  =  19{. 

14.  l=^l,d=ls=-20.  16.  Z  =  -4,,i  =  _^,s  =  _i^. 

How  many  consecutive  terms  must  be  taken  from — 

17.  1,  1^,  2  .  .  .  .  to  make  45? 

18.  },  i,  J  .  .  . .  to  make  - 1  ? 

19.  J^,  2,  f  .  .  .  .  to  make  -20|? 

20.  I,  j,  1  ...  .  to  make  4.5? 

309.  Arithmetical  Means.  If  it  be  required  to  insert  a 
given  number  of  arithmetical  means  between  two  given 
numbers,  the  solution  of  the  problem  is  readily  effected  by 
means  of  Problem  I.  (Art.  307).  The  quantities  in  the  A.  P. 
which  are  given  are  seen  to  be  a,  l,  n,  and  it  is  required  to 
find  d. 


320  ALGEBRA. 

Ex.  Insert  9  arithmetical  means  between  1  and  5. 
We  have  given      a  =  l,    1—5,    n  =  ll. 
Hence,  we  find  ^  =  f  • 

The  required  means  are  therefore  If,  If,  2f ,  .  .  . 


In  case  but  a  single  arithmetical  mean  is  to  be  inserted 
between  two  quantities,  a  and  b,  this  one  mean  is  found  mosi 

readily  by  use  of  the  formula 

For  if  z  denote  the  required  mean,  the  A.  P.  is  a,  x,  b. 
Hence,    x  —  a  —  b  —  x 
2x  =  a  +  b 
a-\-b 


EXERCISE  113. 

Insert — 

1.  Four  arithmetical  means  between  7  and  —  3. 

2.  Seven  arithmetical  means  between  4  and  6. 
8.  Eight  arithmetical  means  between  |  and  3. 

4.  Thirteen  arithmetical  means  between  ^  and  —  §. 
6.  Fifteen  arithmetical  means  between  —  4J  and  9. 

6.  The  arithmetical  mean  between  2f  and  —  5f . 

7.  The  arithmetical  mean  between  x  +  1  and  x  —  1. 

8.  The  A.  M.  between  -  and  -  •     Between  and 


b  ^  x-\-y         x—y 


310.  Miscellaneous  Examples. 

Ex.  1.   The  7th  term  of  an  A.  P.  is  5,  and  the  14th  term  it 
9.  Find  the  first  term. 


ARITHMETICAL  PBOGBESSION,  321 

By  the  use  of  Formula  I.  (Art.  306), 

the  7th  term  is  a  +  6d,        the  14th  term  is  a  +  13d. 

.'.  a  +  ed'=5 (1) 

a  +  13d  =  -  9 (2) 

Subtracting  (1)  from  (2),        7d  =-14 

d=  -2 
Substitute  for  d  in  (1),     a  -  12  -  5 

a  =  17,  Result. 

Ex.  2.   The  sum  of  five  numbers  in  A.  P.  is  15,  and  the 
sum  of  the  first  and  fourth  numbers  is  9.     Find  the  num- 
bers. 
Denote  the  numbers  by 

x  +  2^,     x^y,    X,    x-y,    x-  2y. 

Add,  5a;  =  15 (1) 

Also,  {x  +  2^/)  +  (a;  -  y)  =  9 

.  • .  2a:  +  y  =  9 (2) 

From  (1)  a;  =  3 ;  ,     hence,  from  (2),  y  =  ^. 
Hence,  the  numbers  are  9,  6,  3,  0,  —  3,  Result. 

Similarly,  in  dealing  with  four  unknown  quantities  in  A.  P.,  we  denote 
them  by 

X  +  Sy,    x  +  y,    x-y,    x  -  3y. 

EXERCISE  114. 

Find  the  first  two  terms  of  the  series  wherein — 

1.  The  4th  term  is  11  and  the  10th  is  23. 

2.  The  6th  term  is  —  3  and  the  12th  is  - 12. 

3.  The  7th  term  is  -^  and  the  16th  is  2J. 

4.  The  fifth  term  is  c  —  35  and  the  liih  is  36  —5c. 

5.  Find  the  sum  of  the  first  n  odd  numbers. 

6.  Find  the  sum  of  the  first  n  numbers  divisible  by  7. 

7.  Which  term  in  the  series  IJ,  1^,1^,  .  .  .  •  is  18? 

8.  The  first  term  of  an  arithmetical  progression  is  8;  the 
3d  term  is  to  the  7th  as  the  8th  is  to  the  10th.  Find  the 
series. 

21 


322  ALGEBRA. 

9.  Find  four  numbers  in  A.  P.,  sUch  that  the  sum  of  the 
first  two  is  1,  and  the  sum  of  the  last  two  is  — 19. 

10.  Find  four  numbers  in  A.  P.  whose  sum  is  16  and 
product  is  105. 

11.  A  man  travels  2^  miles  the  first  day,  2|  the  second,  3 
the  third,  and  so  on ;  at  the  end  of  his  journey  he  finds  that 
if  he  had  traveled  6J  miles  every  day  he  would  have  re- 
quired the  same  time.     How  many  days  was  he  walking? 

12.  The  sum  of  10  numbers  in  an  A.  P.  is  145,  and  the  sum 
of  the  fourth  and  ninth  terms  is  5  times  the  third  term.  Find 
the  series. 

13.  If  the  11th  term  is  7  and  the  21st  term  is  8f,  find  the 
41st  term  of  the  same  A.  P. 

14.  In  an  A.  P.  of  21  terms  the  sum  of  the  last  three  terms 
is  23,  and  the  sum  of  the  middle  three  is  5.     Find  the  series. 

15.  Required  five  numbers  in  A.  P.,  such  that  the  sum  of 
the  first,  third,  and  fourth  terms  shall  be  8,  and  the  product 
of  the  second  and  fifth  shall  be  —54. 

16.  The  sum  of  five  numbers  in  A.  P.  is  40,  and  the  sum 
of  their  squares  is  410.     Find  them. 

17.  The  14th  term  of  an  A.  P.  is  38;  the  90th  term  is  152, 
and  the  last  term  is  218.     Find  the  number  of  terms. 

18.  How  many  numbers  of  two  figures  are  there  divisible 
by  3?  By  7  1  How  many  numbers  of  three  figures  are 
divisible  by  6  ?     By  9  ? 

19.  How  many  numbers  of  four  figures  are  there  divisible 
by  11  ?  Find  the  sum  of  all  the  numbers  of  three  figures 
divisible  by  7. 

20.  If  a  body  falls  16 A  ft.  the  first  second  of  its 
fall ;  three  times  this  distance  the  second ;  five  times  the 
third,  and  so  on,  how  far  will  it  fall  the  30th  second  ? 
How  far  will  it  have  fallen  during  the  30  seconds  ! 

21.  If  a,  5,  c,cZ  are  in  A.  P.  prove:  {l)thsita-\-d  =  h-{-c: 
(2)  that  ale,  hJc,  ck,  dJc  are  also  in  A.  P. ;  and  (3)  that  a-\-Jcj 
J)  -{-  k,  c  -{-  Jc,  d  -\-  k  are  in  A.  P.  State  this  problem  with- 
out the  use  of  the  symbols,  a^h^c^  d,  k. 


CHAPTER   XXV. 

GEOMETRICAL  AND  HARMONICAL  PROGRES- 
SIONS. 

311.  A  Geometrical  Progression  is  a  series  each  term  of 
which  is  formed  by  multiplying  the  preceding  term  by  a  con- 
stant quantity  called  the  ratio. 

Thus,  1,  3,  9,  27,  81, is  a  geometrical  progression 

(or  G.  P.)  in  which  the  ratio  is  3. 

Given  a  geometrical  progression,  to  determine  the  ratio: 
divide  any  term  by  the  preceding  term. 


Thus,  in  the  G.  P.,    -  3,  |,  -  }, 


the  ratio  —  -^  =  —  i« 

—  o 

312.  Quantities  and  S3nnbols  Used,  a,  ?,  n,  s  are  used, 
as  in  A.  P.    Besides  these,  r  is  used  to  denote  the  ratio. 

313.  Two  Fundamental  Formulas.  Since  in  a  G.  P.  each 
term  is  formed  by  multiplying  the  preceding  telm  by  the 
common  ratio,  r,  the  general  form  of  a  G.  P.  is — 


a,    ary    ar',    ar^y    ar\ 


Hence,  the  exponent  of  r  in  each  term  is  one  less  than  the 
number  of  the  term.    Thus, 

The  10th  term  is  aT^, 

■     The  15th  term  is  ar'\ 

The  nth  term,  or  l  =  af"* (1) 

323 


324  ALGEBRA. 

'   In  deriving  a  formula  for  the  sum,  we  know,  also, 

s  =  a  ^- ar  ^- ar^  r\. _|-  ^^.n- 1  .    ^    ,  (2) 

Multiply  (2)  by  r, 

rs  ^  ar  +  ar^ -[- ar^ -{■ +  af'^  +  af  .  (3) 

Subtract  (2)  from  v3^, 

rs  —  s  =  af  —  a. 

ar^  —  a  '  ^.^ 

•••  s^ T (^) 

r  — 1 

Multiply  (1)  by  r, 

Substitute  rl  for  ^r"*  in  (4), 

rl—a  ,-. 

s== r (5) 

r  — 1 

Hence,  collecting  the  results  obtained  in  (1),  (4),  (5),  we 
have  the  two  fundamental  formulas  for  I  and  s: 

I.  l^ar^'-K 

11.  5- - 

r  —  1 

W  — a 

Ex.  1.   Find  the  8th  term  and  sum  of  8  terms  of  the  G.  P., 

1,  3,  9,  27 

.  In  this  case,      a  =  1,     r  =  3,     n  =  8. 
From  I.,  ^  =  1x3'  =  2187. 

From  II.,  s  =  ^  ""  ^^^^  ~  ^  =  3280. 

3-1 

Ex.  2.   Find  the  10th  term  and  the  sum  of  10  terms  of  the 
G.P.,  4,  -2,  1,  -i 

Here      a  =  4,    r  =  —  ^,     n  =  10. 
Hence,     I  -  4{- If  ^  -  ^U  =  -  jh- 

8  =  (-i)(-Th)-^  ==  341. 

—  J  —  1 


GEOMETRICAL  PROGRESSION,  326 

EXERCISE  115. 

1.  Find  the  sixth  term  in  the  series  2,  6,  18, 

2.  Find  the  7th  term  in  3,  6,  12, 

3.  Find  the  6th  and  the  sum  of  6  terms  in  45,  — 15, 5, 

4.  Find  the  5th  and  the  sum  of  5  terms  in  81,  —  54, 

5.  Find  the  7th  and  the  sum  of  7  terms  in  1|,  —  |, 

6.  Find  the  9th  term  in  the  series  2,  2l/2,  4, 

Find  the  sum  of  the  series — 

7.  3,  -  6,  12, ....  to  6  terms. 

8.  27,  -  18,  12, ....  to  7  terms. 
>    9.  —f,  1^,  —  2, ....  to  9  terms. 

10.  h-h-h,'  •  •  •  to  8  terms. 

11.  J  1,  V^, ....  to  8  terms. 

12.  V^,  1/6,  2V3, ....  to  10  terms. 

13.  V^  —  1,  1,  1/2  +  1,  ....  to  6  terms. 

B14.  Problem  I.  Given  three  of  the  five  quantities,  a,  I,  n, 
s   r,  to  determine  the  other  two. 

As  in  A.  P.,  in  the  two  fundamental  formulas  (I.  and  II., 
A-'t.  313)  substitute  for  the  three  known  quantities,  and  de- 
termine the  other  two  quantities  by  solving  the  resulting 
equations. 

Ex.  1.  Given  a=-2,  n  =  7,  l  =  -12S;  find  r,  s. 

From  I.,  -  128  =  -  2r«. 

Hence,  r«  =  64,        r  =  ±  2. 

Frora  II,  if  r  =  +  2,  s  =  ^(-m_-^{-^)  =  -  256  +  2  =  -  254. 

I,,_2,  ,^(-2)(-128)-(-2)^25^_3,^ 

HenotJ,  there  are  two  sets  of  answers ;  viz.,    r  =  +  2,         8  =  -  254. 

r  =  ~  2,        «  =  -  86. 


326 


ALGEBRA, 


No. 


Given. 


Required. 


Formulas. 


9 
10 
11 
12 
13 
14 
15 
16 

17 
18 
19 
20 


a,  r,  s 

r,  n,  s 
a,  n,  s 


r,  n,  I 


r,  I,  s 
n,  I,  8 


a,  r,  n 
a,  r,  I 
a,  n,  I 


a,  n,  I 
a,  n,  s 
a,  I,  s 


a,  r,  I 
a,  r,  s 
a,  I,  8 
r,  I,  s 


a  +  (r  —  1)8 

r 
(r  —  l)sr^-* 

r "  - 1 
-  ^)"-i  -  a{8-  ay 


a  -- 

a{i 


(r  -  1)8 
r^-1  ' 
rl—  {r  -  l)s. 


i{s  -  ly 


a{7^- 

1) 

Ir  —  a 
r-1 

L 

n 

-^a» 

"-^r- 

(/•"- 

n  - 

1)1 

-fa 

(r  -  l)r»*- 


^a 


a  a 

8  —  a 

8-1  ' 

«   .r»-'  +  — ^  =0. 


s-  I 


s-l 


log  ^  -  log  «  _^  -|^ 

logr 
log  [g  +  (r  —  1  )s1  —  log  g 
logr 

log  I-  log  g ^  J 

log  (s  -  g)  -  log  (s-l) 
log^-  log[^r-  (r-  l)s1  ^  ^ 
logr 


GEOMETRICAL  PROGRESSION,  327 

Ex.  2.  Given  a-:f,  r  =  -J,  8  =  ^\;  find  I,  n. 
From  I.,  ^=3(_|)n_i (Ij 

Fromll,  ^-i^  -  ^^"1^2  1^ (2) 

Whence  iff  =  (~y~^ 

Whence  n  =  6. 

Substitute  for  w  in  (1),       I  =  —  ^^^j. 

315.  Problem  II.  Given  three  of  the  five  quantities^  a,  I,  n, 
Sf  r,  to  obtain  a  formula  for  one  or  both  of  the  other  two  in  terms 
of  the  three  given  quantities. 

Ex.   Given  n,  r,  s,  obtain  a  formula  for  I. 

Using  l  =  ar''-^ (1) 

»=.75f (2) 

Solve  (2)  for  a,  a  =  rl  -  s{r  -  1)  .   .   . (3) 

Substitute  in  (1)  for  a  from  (3), 

I  =  rH  —  sr'^  -  \r  —  1). 
Hence,  (r*  -  \)l  =  sr'^-^ir  —  1) 

I  =  sr^-'^jr  -  1) 
r«-  1 
A  complete  table  of  all  possible  formulas  for  G.  P.  is  given  on  the  oppo- 
site page.   These,  the  student  should  be  required  to  derive  for  himself  (except 
those  for  n). 

EXERCISE  116. 

Find  the  first  term  and  the  sum  when — 

1.  n  =  6,r  =  3,  Z  =  486.  4.  7i  =  8,r  =  -|,Z=  -ffj. 

2.  71  =  8,  r  = -2,/=  -640.         6.  n  =  9,  r= -3, /= -1215. 

3.  n  =  7,r  =  f, /  =  if^.  6.  n  =  7,  r  =^1/6, /  =  3. 

Find  the  ratio  when — 

7.  a=-2,  ^  =  2048,  n  =  6.         9.  a  =  2|f,  Z=  -  Jf,  n  =  6. 

8.  a  =  9,/  =  2^,5  =  23|.  10.  a=-16i,^=^,5=-i2A. 


328  ALGEBRA. 

Find  the  number  of  terms  when — 

11.  a=^^,l=  iV,  ^  =  i  13.  a  =  18,  r  =  -  §,  s  =  12f . 

12.  a  =  3,Z=-96,s--63.        14.  Z= -8,r= -2,s= -5^. 

How  many  consecutive  terms  must  be  taken  from  the 
series — 

15.  i,i,i, tomakefi^? 

16.  15|,  —61  2i  ....  to  make  10|? 

17.  5^,  —8,  12, ....  to  make  -221? 

316.  Geometrical  Means.  If  it  be  required  to  insert  a 
given  number  of  geometrical  means  between  two  given 
numbers,  the  solution  of  the  problem  is  readily  effected 
by  means  of  Problem  I.  (Art.  314).  The  quantities  whicn 
are  given  are  seen  to  be  a,  /,  n,  and  it  is  required  id 
find  r. 

Ex.  Insert  5  geometrical  means  between  3  and  ^^. 

We  have  given  a  =  3,  1  =  ^^,  n  =  l,  to  find  r. 

Solving  by  Problem  I.,      r  =  \. 

Hence,  the  required  geometrical  means  are, 

In  case  but  one  geometrical  mean  is  to  be  inserted  be- 
tween two  given  quantities,  a  and  6,  this  one  mean  is  found 
most  readily  by  using  the  formula  Vah.  For  if  x  rep- 
resent the  geometrical  mean  between  a  and  6,  the  series 
will  be 

a,    Xy    b, 

X      h 
Hence,   -  =  - . 
a      X 

,' ,  x^  =  ah. 

X  =  VaS, 


GEOMETRICAL  PROGRESSION,  329 

EXERCISE  117. 

Insert — 

1.  Three  geometrical  means  between  8  and  ^. 

2.  Three  geometrical  means  between  J  and  |. 

3.  Six  geometrical  means  between  ^V  and  —  J/-. 

4.  Four  geometrical  means  between  —  |  and  3584. 
6.  Six  geometrical  means  between  56  and  —  -^. 

6.  Five  geometrical  means  between  f  and  12. 

Find  the  geometrical  mean  between — 

7.  4i  and  |.  8.  3f  and  ^. 

9.  51/2  +  1  and  51/2-1. 
10.  31/5  + 2l/e  and  3l/o- 21/3. 

11.  28a'a;and  63aV"  12.  %-^  and  ^    ^  . 

c'v^  a:  1/^5- 

a'  8 

13.  Insert  6  geometrical  means  between  —  and 

lb  Y^ 

8  n^ 

14.  Insert  7  geometrical  means  between  —  and  — 

n^  2 

317.  Problem  III.     To  find  the  limit  of  the  sum  of  an  infinite 
decreasing  geometrical  progression. 

If  a  line  AB 

C  D 

A  I B 

1  111 

2  T  "g-     TS" 

be  given  of  unit  length,  and  one-half  of  it  (AC)  be  taken, 
and  then  one-half  of  the  remainder  (CD),  and  one-half  of 
the  remainder,  and  so  on,  the  sum  of  the  parts  taken  will  be 

i  +  i  +  i  +  A  +  A+ 


330  ALGEBRA. 

This  is  an  infinite  decreasing  G.  P.  in  which  r  =  ^.  But 
the  sum  of  all  these  parts  must  be  less  than  1,  but  approach 
closer  and  closer  to  1  as  a  limit,  the  greater  the  number  of 
parts  taken.  This  is  an  illustration  of  the  meaning  of  the 
limit  of  an  infinite  decreasing  G.  P. 

In  general,  to  find  the  limit  of  an  infinite  decreasing  G.  P. 
we  have  the  formula 


III. 


a  —  VL 

Formula  II.  of  Art.  313  may  be  written,  s  — ■-     Then, 

1  —  r 
as  the  number  of  terms  increases, 

I  approaches  indefinitely  to  0. 

rl 

.'.a-rl 

a  —  rl  ^^ 

\—r  1  — r 


u 

0. 

(( 

a-0  =  a. 

a 

a 

1-r 
Ex.  Find  the  sum  of  9,  —  3,  1,  —J,  .  .  .  .  to  infinity. 
Here  a  =  9,    r=—\. 

318.  Repeating  Decimals.  By  the  use  of  the  Formula 
III.  of  Art.  317,  the  value  of  repeating  decimals  may  be 
determined. 

Ex.  1.   Find  the  value  of  0.373737 

0.373737 =  .37  +  .0037  +  .000037  + 

Here  a  =  .37,    r  =  .01. 
.  .37     _  .37  _^ 

•   •'~1-.01~.99~^- 


GEOMETRICAL  PROGRESSION,  331 

Ex.  2.   Find  the  value  of  3.1186186 

3.1186186  -  3.1  +  .0186  +  .0000186  -f 

Setting  aside  3.1,  and  treating  the  remaining  terms  as 
a  G.  P., 

a  =  .0186,        r  =  .001. 

1~.001       .999  ~^^^--sih' 
•'•3.1186186 =3A  +  rf|^  =  3^. 


EXERCISE  118. 

Find  the  sum  to  infinity  of  the  series — 

l-2,i|, 6.  A,A,^,... 

2.  2,-1,  i 7.  2if,~lJ,li, 

3.  - 9,  6,  -4, 8.  6,  SV2,  3, .  . 


4.  l|,li,f, '        ^         1  ,         1 


6.4J,-2i,li, 1/2-1       '   1/2  +  1 

10.  il/2  +  ii/3  +  ii/2". 

■■•■+('-^('-jj+ 

Find  the  values  of— 

12.  0.63.  13.  0.417.  14.  5.846. 

15.  3.52424 18.  1.02727 

16.  1.4037037  .....  19.  1.027027 

17.  3.215454 20.  0.30102102 


819.  Miscellaneous  Problems. 

Ex.  Find   four  members   in  G.  P.,  such   that  the    sum 


332  alqebha, 

of  the  first  and  fourth  is  56,  and  of  the  second  and  third 
is  24. 
Denote  the  required  numbers  by  a,  ar,  ar\  ar^. 
Then  *  a  +  a?'^  -  56 

ar  +  ar"^  =  24. 

Or,  a(l  +  r')  =  56 (1) 

aril  +  r)  =  24 (2) 

Divide  (1)  by  (2),         ^  ~  ^  "^  ^'  - 1. 
r 

Hence,  3  -  3r  +  Sr^  =  7r. 

3r2  -  lOr  =  -  3 

r  =  3,     or  |. 

And  a  =  2,     or  54. 

Hence,  the  numbers  are,         2,  6,  18,  54. 

Or,  54,  18,  6,  2. 

EXERCISE  119. 

Find  the  first  two  terms  of  the  G.  P.  wherein^ 

1.  The  3d  term  is  2,  and  the  5th  is  18. 

2.  The  4th  term  is  f  and  the  9th  is  48. 
8.  The  3d  term  is  5  and  the  8th  is  ^K 
4.  The  5th  term  is  6  and  the  11th  is  -A-. 


Determine  the  nature,  whether  Ar.  or  Geom.,  of  each — 

^'  hhi 8.  I,  f ,  |- 

6.  i,  i,  i 9.  3i,  41,  7i 

T-iiiV 10.  7i5f,4T% 


11.  Divide  65  into  3  parts  in  geometrical  progression,  such 
that  the  sum  of  the  first  and  third  is  3^  times  the  second 
part. 

12.  There  are  3  numbers  in  G.  P.  whose  sum  is  49,  and  tho 
sum  of  the  first  and  second  is  to  the  sum  of  the  first  and  third 
as  3  to  5.     Find  them. 


HARMONICAL  PROGRESSION.  333 

13.  The  sum  of  three  numbers  in  G.  P.  is  21,  and  the  sum 
of  their  reciprocals  is  -^.     Find  them. 

14.  Find  four  numbers  in  G.  P.,  such  that  the  sum  of  the 
first  and  third  is  10,  and  of  the  second  and  fourth  is  30. 

15.  Three  numbers  whose  sum  is  24  are  in  A.  P.,  but  if  3, 
4,  and  7  be  added  to  them  respectively,  these  sums  will  be  in 
G.  P.     Find  the  numbers. 

16.  The  sum  of  $225  was  divided  among  four  persons  in 
such  a  manner  that  the  shares  were  in  G.  P.,  and  the  differ- 
ence between  the  greatest  and  least  was  to  the  difference 
between  the  means  as  7  is  to  2.     Find  each  share. 

1  2 

17.  Find  the  sum  of  — ,  i/2   -^= .  ...  ad  infinir 

1/2-1'  "^    '   1/2  +  1 
turn. 

18.  There  are  4  numbers  the  first  3  of  which  are  in  G.  P., 
and  the  last  3  are  in  A.  P, ;  the  sum  of  the  first  and  last  is  14, 
and  of  the  means  is  12.     Find  them. 

19.  If  the  series  f ,  ^  .  .'  .  .  be  arithmetical,  find  the  102d 
term;   if  geometrical,  find  the  sum  to  infinity. 

20.  Divide  $369  among  A,  B,  C,  and  D  so  that  their  shares 
may  be  in  G.  P.,  and  the  sum  of  A's  and  B's  shares  shall  be 
$144. 

21.  The  ages  of  three  men,  A,  B,  and  C,  are  in  G.  P.,  and 
the  sum  of  their  years  is  111 ;  but  |  of  A's  age,  -^  of  B's,  and 
^  of  C's  form  an  A.  P.     Find  their  ages. 

22.  Insert  between  2  and  9  two  numbers,  such  that  the  first 
three  of  the  four  may  be  in  A.  P.  and  the  last  three  in  G.  P. 

23.  Prove  that  the  series  l/2-l,  31/2-4,  2(5 1/2-7) 
....  is  geometrical ;  that  its  ratio  is  2  —  l/2 ;  and  that  its 
sum  to  infinity  is  unity. 

HARMONICAL  PROGRESSION. 

320.  An  Harmonical  Progression  is  a  series  the  recipro- 
cals of  whose  terms  form  an  arithmetical  progression. 

Thus,  1,  :j,  7,  iVj is  an  harmonical  progre^ion,  sine©- 

the^?^biprocals  of  its  terms, 


334  ALQEBBA. 

1,  4,  7,  10 

form  an  arithmetical  progression. 

The  general  form  of  an  harmonical  progression  is — 

1111  1 

, }    ~  1 


a      a-\-  d       a-\r2d      a-\-Zd  a-\-  {n  —  V)d 

321.  General  Principle.  Problems  relating  to  harmonical 
progression  are  solved  to  best  advantage  by  taking  the  recip- 
rocals of  the  terms  of  the  given  progression,  and  solving  the 
A.  P.  thus  formed. 

Ex.  1.   Find  the  15th  term  of  the  H.  P.,  hhhii 

Taking  the  reciprocals  of  the  terms  of  the  given  series,  we 
obtain  the  A.  P., 

2,  5,  8,  11 

In  this  a  =  2,     c2  =--^  3,    n  =  15. 

.-.  Z  =  2  +  l4X3=--44. 
Hence,  the  15th  term  of  the  given  H.  P.  is  the  reciprocal  of 
44 ;  that  is,  4^. 

Ex.  2.   Insert  7  harmonic  means  between  —  3  and  4. 

To  solve  this  problem  it  is  necessary  to  insert  7  arithmetical  means  be- 
tween —  \  and  \. 

Hence,  «  =  —  i>     ^  ^  h     ^  =  9« 

,' .  \=  -  \  +  M,     d  =  ^. 
Inserting  arithmetical  means,  the  A.  P.  is 

ri>    ~  ij?j    sVj    ts>    hh    h 

-  3,    -  If,  -  V,    -  ff,   -  24,   32,   -V-,   f f,  4. 

322.  Harmonic  Mean.  If  but  one  harmonic  mean  is  to 
be  inserted  between  a  and  b,  and  this  be  denoted  by  if,  then 

-—  is  the  arithmetical  mean  between  —  and  ^ :  and  by  Art.  309, 


HARMONICAL  PEOOBESSION.  336 


1        a      b        a  +  b  „       2ab 

£1  = 


H  2  2ab  a  +  b 

If  the  arithmetical,  geometrical,  and  harmonic  means  be- 
tween a  and  b  be  denoted  by  A,  (r,  H^  respectively, 

we  have        A  =  — - — >     G  =  VoBj    H 


2  '  a  +  b 

,'.  H=abX-^  =  G'X^' 
a  +  b  A 

.'.  G'  =  AH  and  G=--VAB. 

Hence,  the  geometrical  mean  between  two  quantities  is  also  the 
geometrical  mean  between  their  arithmetical  and  harmonic  means, 

EXERCISE  120. 

Find  the  last  term  in  the  series— 

1-  I)  T>  4)  •  •  •  •  to  20  terms. 

2.  f ,  2,  6, .  .  . ..  to  18  terms. 

3.  -j%,  -|,  -1, ....  to  27  terms. 

Insert— 

4.  Four  harmonic  means  between  J  and  |-. 

5.  Five  harmonic  means  between  f  and  ■^. 

6.  Seven  harmonic  means  between  —  3^  and  —  ^. 

Find  the  harmonic  mean  between — 

7.  2|  and  3.  8.  4i  and  -  3|.  9.  ^  and  ^. 

^  ^  ^  x-\-l  x-1 

10.  The  first  term  of  a  H.  P.  is  x,  and  the  second  term  is  y. 
Find  the  next  two  terms. 

11.  The  arithmetical  mean  between  two  numbers  is  15,  and 
the  H.  M.  is  14|.     Find  the  numbers. 

12.  The  fifth  term  of  a  H.  P.  is  - 1,  and  the  14th,  is  f    Find 
the  series. 

13.  The  G.  M.  between  two  numbers  is  16,  and  the  H.  M. 
is  12|.     Find  the  numbers. 


CHAPTER    XXVI. 
PERMUTATIONS  AND  COMBINATIONS. 

323.  The  Permutations  of  a  group  of  objects  are  the  differ- 
ent arrangements  which  can  be  made  of  the  objects  with  respect 
to  their  order. 

Thus,  the  permutations  of  the  three  letters  a,  6,  c  are,  abc, 
acb,  bac,  boa,  cab,  cba.  Taken  two  letters  at  a  time,  the  per- 
mutations of  a,  6,  c  are,  a6,  6a,  ac,  ca,  be,  cb. 

324.  The  Combinations  of  a  group  of  objects  are  the  dif- 
ferent collections  which  can  be  made  from  them  without  respect 
to  order. 

Thus,  the  combinations  which  can  be  made  from  three  let- 
ters, a,  6,  c,  taken  two  at  a  time,  are,  ab,  ac,  be.  It  is  seen 
that  ba  and  ab,  for  instance,  are  different  permutations,  but 
are  the  same  combination. 

325.  The  Number  of  Permutations  in  a  Group  of  ft  Ele- 
ments: To  determine  the  number  of  permutations  that  can 
be  made  of  a  group  of  objects  by  actually  writing  out  the 
permutations  and  counting  them  usually  involves  so  much 
labor  as  to  make  the  method  impracticable.  Instead  of  this, 
it  is  possible  to  establish  a  formula  by  which  the  number  of 
permutations  in  a  group  of  objects  can  be  determined  by  a 
simple  process  of  multiplication  from  the  number  of  objects 
or  elements  in  the  group. 

Thus,  to  determine  the  number  of  permutations  which  can 
be  made  with  a  group  of  four  objects,  as  a,  6,  c,  d,  we  conceive 
the  permutations  to  be  formed  b}^  writing  each  of  the  four 
elements  in  the  first  place  in  turn ;  after  each  of  which  the 
remaining  three  elements  may  be  placed;  after  each  of  which 
the  remaining  two,  and  so  on, 

336 


PERMUTATIONS  AND  COMBINATIONS. 


337 


Thus,  we  obtain 
[cd 

[hd 
Xdb 
{he 


d 


cb 


ad 
da 
ac 
ca 


M 


c  < 


Whence  we  have  all  the  permutations, 
abed,  abde,  etc.;  as,  4  X  3  X  2  X  1,  or  24  in 
number. 

Similarly,  to  form  the  number  of  permu- 
tations which  can  be  made  with  a  group  of 
n  elements,  we  may  write  each  of  the  n  ele- 
ments in  the  first  place;  after  each  of 
which  the  remaining  n  —  1  elements  may 
be  written;  after  each  of  which  the  re- 
maining 71  —  2  elements,  and  so  on. 

Hence,  the  number  of  permutations  formed 
from  a  group  of  n  elements  is 

nCn  — l)(n-2)  .  .  .  .  8X2X1. 

If  the  permutations  of  four  objects  be 
formed,  taking  two  objects  at  a  time,  in  the 
first  place  each  of  ^the  four  objects  may  be 
written,  after  each  of  which  the  remaining 
three,  which  exhausts  the  number  of  per- 
mutations so  formed.  Hence,  the  permu- 
tations of  four  objects,  taken  two  at  a  time, 
is  4  X  3. 

Similarly,  the  permutations  of  n  objects 
taken  2  at  a  time  is  n(7i  —  1)  ;  of  n  objects 
taken  r  at  a  time,  is 

n(n-l)(n-2) (n-r  +  l); 

[r  factors  in  all]. 

326.  Symbols  Used.   For  the  number  of  permutations  of  a 
group  of  n  objects,  taken  r  at  a  time,  the  symbol,  „P^,  is  used. 
The  product  n(n  —  1)  ....  3  X  2  X  1   is   called  factorial  w, 
and  is  abbreviated  into  the  form,  |n,  or  ?il. 
Hence,  from  the  preceding  Article, 

nP„  =  n(7i-l)(n-2)  ....  3X2Xl=|ri,ornI. 
nPr  =  n{n-l)(n-2)  ....  (n-r  +  l). 

22 


db 
ad 
da 
ab 
ba 

be 
cb 
ae 
ca 
ab 
ba 

4X3X2X1 


338  ALGEBRA, 

Ex.  1.  In  how  many  ways  may  the  letters  which  form  the 
word  Baltimore  be  arranged  ? 

Since  there  are  9  different  letters  in  the  word  Baltimore,  we 
have 

9P9  =  9X8X7X6X5X4X3X2X1  =  362,880  Permutations. 

Ex.  2.  How  many  of  the  permutations  in  Ex.  1  will  begin 
with  the  letter  I  ? 

Since  the  letter  I  remains  fixed  in  the  first  place,  8  letters 
are  arranged  in  different  orders.     Hence,  we  have 
gPs  =  [8  =  40,320  Permutations. 

327.  The  Number  of  Combinations  in  a  Group  of  n  Ele- 
ments. We  denote  the  number  of  combinations  which  can  be 
made  with  a  group  of  n  elements,  taken  r  at  a  time,  by  „Cv. 

The  permutations  of  a  group  of  n  objects,  taken  r  at  a  time, 
might  be  formed  by  writing  all  the  different  combinations  of 
the  n  objects  taken  r  at  a  time,  and  then  making  all  possible 
permutations  (that  is,  arrangements)  of  the  elements  of  each 
combination  of  r  objects;  viz.  |r_ arrangements. 

Thus,  for  example, 

Or,  in  general, 

„p,=„ax|r 

p 

[r 
^ n(n-l)(n-2)  ....  (n-r  +  1)^ 
"  *■"  r(r-l)(r-2)  ....  1 

Ex.  How  many  combinations  can  be  formed  with  the  letters 
of  the  word  Baltimore^  taken  four  at  a  time  ? 

We  have 

^      9X8X7X6      ,^^^     ,.      . 

•Ci  = =  126  Combinations, 

4X3X2X1 


PERMUTATIONS  AND  COMBINATIONS.  339 

828.  Other  Formulas  for  „c;.  The  formula  for  ^C^  may  be 
put  into  another  and  often  more  convenient  form  by  multi- 
plying both  numerator  and  denominator  by  \n  —  r. 

n(n  — 1)  .  .  .  .  (n  — r  +  1)  In  —  r  In 

Thus,  „a= '——T ^i-==^_. 

\r\n  —  r  \r\n  —  r 

It  is  also  to  be  observed  that  for  each  different  selection  (or 
combination)  taken  from  a  group  of  objects  a  different  group 
is  left.  Thus,  if  from  a,  6,  c,  d^  e  we  take  a,  6,  cZ,  we  have  left 
c,  e ;  if  we  take  6,  c,  e,  we  have  left  a,  d,  and  so  on. 

Hence,  s^s^sC^s 

Or,  in  general,  nCr  =  nCn  -  r- 

Ex.  Determine  the  number  of  combinations  of  20  letters 
taken  18  at  a  time. 


20^18  —  20^2 

^  20  X 19^ 
1X2 


=  190  Combinations, 


EXERCISE  121. 

Find  the  values  of— 

1.  6^4-  3.  ^Ps'  5.  19C16.  7.  jiPfi. 

2.  20C9.  4.  uO,.  6.  uP*-  8.  M. 

9.  How  many  different  numbers  of  5  different  figures  each 
can  be  formed  from  the  nine  significant  digits? 

10.  From  a  company  of  24  men,  in  how  many  ways  can  a 
committee  of  10  be  selected? 

11.  How  many  words  of  5  different  letters  can  be  formed 
from  our  alphabet? 

12.  There  are  13  points  in  a  plane,  no  three  of  which  are  in 
the  snme  straight  line.  How  many  different  triangles  can  be 
formed  having  three  of  the  points  for  vertices  ? 


340  ALGEBRA. 

13.  How  many  different  words  of  eight  letters  each  can  be 
formed  from  the  letters  in  the  word  republic  f 

14.  How  many  different  numbers  of  4  different  figures  can 
be  formed  from  the  ten  digits  0,  1,  2, ? 

15.  From  the  letters  in  the  word  universal  how  many  words 
can  be  formed,  taking  6  at  a  time  ?     Taking  7  at  a  time  ? 

16.  Five  persons  enter  a  car  in  which  there  are  seven  seats. 
In  how  many  ways  may  they  take  their  places  ? 

17.  How  many  different  throws  can  be  made  with  2  dice  ? 

18.  How  many  different  throws  can  be  made  with  3  dice  ? 

19.  How  many  numbers  is  it  possible  to  write  by  use  of  our 
10  digits  without  repeating  any  digit  ? 

SPECIAL  CASES. 

329.     I.  Elements  of  a  Group  repeated.     It  may  happen 

that  some  of  the  elements  in  a  group  of  objects  are  the  same. 

Thus,  if  it  be  required  to  determine  the  number  of  permu 

tations  which  may  be  made  of  the  letters  in  the  word  Chicago^ 

it  is  observed  that  in  this  word  the  letter  c  occurs  twice, 

hence,  all  the  permutations  formed  by  interchanging  the  c's 

are  identical.     Hence,  for  the  total  number  of  permutatiopb 

we  have 

\1 
- —  =  2520  Permutations. 

\i 

In  general,  if  in  a  group  of  n  elements,  one  set  of  elements, 
i  in  number,  are  identical,  and  another  set,  k  in  number,  ar© 
identical,  the  total  number  of  permutations  in  the  entire  group, 
taken  all  together,  is 

\n 


\i\h 

Ex.  How  many  permutations  can  be  formed  from  the  let* 
ters  of  United  States,  taken  all  together? 

We  have  12  letters  in  all,  but  t  used  three  times,  s  used 
twice,  and  e  twice. 


PERMUTATIONS  AND  COMBINATIONS.  341 

Hence,        — ■ — — -  =  19,958,400  Permutation. 

[£[2  1-? 


II.  Different  Groups  taken  Together.  If  combina- 
tions be  formed  from  one  group  of  elements,  and  other  combina- 
tions from  another  group  of  elements,  and  these  combinations 
be  taken  together  in  pairs,  one  from  each  group,  it  is  evident 
that  the  total  number  of  combinations  will  be  the  product  of 
the  number  of  combinations  in  one  group  by  the  number  in 
the  other  group.  For  each  combination  in  the  first  group 
may  be  joined  separately  to  each  combination  of  the  second 
group. 

Ex.  Out  of  15  Republicans  and  10  Democrats,  how  many 
different  committees  may  be  formed  of  4  Republicans  and  3 
Democrats  ? 

We  have  for  the  number  of  partial  committees  from  the 
Republicans, 


.       15-X  14  X  13  X  12      ,_^ 
4X3X2X1 


From  Democrats, 


10X9X1^,20, 
'"  '        3X2X1 

Hence,  the  number  of  entire  committees  will  be 

1365  X  120  =  163,800,  Result. 


EXERCISE  122. 

Find  the  number  of  different  permutations  that  can  be 
made  from  each  of  the  following  words,  taking  all  the  let- 
ters: 

1.  Inning.  3.  Upper  House.      5.  Independence. 

2.  Successes.      4.  Mississippi.         6.  Unconsciousness. 

7.  How  many  different  arrangements  can  be  made  from 
dbYy  when  written  in  the  expanded  form? 


342  ALGEBRA. 

8.  How  many  committees  of  2  teachers  and  3  boys  can  be 
selected  from  a  school  of  5  teachers  and  25  boys? 

9.  From  9  red  balls,  5  white  balls,  and  4  black  balls,  how 
many  different  combinations  can  be  formed,  each  consisting 
of  5  red,  3  white,  and  2  black  balls  ? 

10.  From  9  merchants,  14  lawyers,  and  7  teachers,  how 
many  different  companies  can  be  formed,  each  consisting  of 
3  merchants,  5  lawyers,  and  2  teachers? 

11.  How  many  different  words,  each  consisting  of  3  con- 
sonants and  1  vowel,  can  be  formed  from  12  consonants  and 

3  vowels? 

Hint.  The  number  of  possible  combinations  is  i^Og  x  gCj.  But  each 
combination  contains  4  letters,  and  may  be  arranged  into  [^  different  per- 
mutations. 

12.  How  many  different  words,  each  consisting  of  4  con- 
sonants and  2  vowels,  can  be  formed  from  8  consonants  and 

4  vowels? 

13.  Find  the  number  of  different  words  of  3  consonants 
and  2  vowels  each,  that  can  be  formed  from  an  alphabet  of  5 
vowels  and  21  consonants. 

14.  In  how  many  ways  can  2  ladies  and  2  gentlemen  be 
chosen  to  make  a  set  at  lawn  tennis  from  a  company  of  6 
ladies  and  8  gentlemen? 

How  many  different  words  can  be  formed  from  the  letters 
in  the  following  words,  using  all  the  letters  in  each  instance : 

15.  Volume  J  the  second,  fourth,  and  sixth  letters  being  vow- 
els? 

16.  Numerical,  the  even  places  always  to  be  occupied  by 
vowels  ? 

17.  Absolute,  the  first  and  last  letters  to  be  vowels? 

18.  Parallel,  the  first  and  last  letters  to  be  consonants  ? 

19.  How  many  different  quadrilaterals  can  be  formed  from 
20  points  in  a  plane,  as  the  vertices,  if  no  three  are  in  the 
same  straight  line? 


FERMUTATIONS  AND  COMBINATIONS.  343 

20.  A  man  has  7  pairs  of  trousers,  5  vests,  and  6  coats.  In 
how  many  different  costumes  may  he  appear? 

21.  In  how  many  different  ways  can  a  base-ball  nine  be 
arranged,  the  pitcher  and  catcher  being  always  the  same,  but 
the  others  playing  in  any  other  position  ? 

22.  How  many  different  sums  of  money  can  be  obtained 
from  a  cent,  a  five-cent  piece,  a  dime,  a  quarter-dollar,  and  a 
half-dollar? 

23.  From  5  labials,  4  palatals,  and  6  vowels  how  many 
words  can  be  formed,  each  containing  2  vowels,  2  labials, 
and  a  palatal? 

24.  There  are  12  persons  in  a  coach  at  the  time  of  an  acci- 
dent at  which  3  are  killed,  4  are  wounded,  and  5  escape  unin- 
jured.    In  how  many  ways  might  all  this  happen  ? 

25.  The  number  of  committees  of  three  members  each 
which  can  be  formed  from  a  certain  number  of  boys,  is  to 
the  number  of  similar  committees  possible  if  there  were  one 
more  boy,  as  10  to  11.     Required  the  number  of  boys. 

26.  The  number  of  words  of  four  letters  each  which  can 
be  made  from  a  certain  number  of  letters,  is  2V  ^^  ^^e  num- 
ber of  words  of  five  letters  each  which  could  be  made  if 
there  were  two  more  letters.  Required  the  number  of  let- 
ters.    Verify  both  results. 

27.  If„C3:2«a=--7:l5,  findn. 

28.  If  20a  =  .0  a +  2,  find, P4. 

29.  Show  that  „  +  iC-  =  nC  +  nCr^X. 

30.  In  how  many  ways  can  nine  children  form  a  ring,  all 
facing  the  centre  ? 


.       CHAPTER    XXVII. 

UNDETERMINED  COEFFICIENTS. 

331.  Convergency  of  Series.    Using  the  jesult  obtained  in 
Art.  317,  viz., 

a 

s  =  - , 

1  —r 

it  is  found  that  the  infinite  series 

i+i+i+i+ (1) 

approaches  a  limit,  =2. 

Similarly,  if  we  take  the  infinite  series 


(2), 


we  find  that  for  any  value  of  x  between  0  and  1,  as  -,  where 

6  <  a,  the  series  has  a  definite  limit ;  viz., 

1 ^      a 

b         a  —  h 
a 

But  if  in  series  (2)  we  let  a;  =  1,  the  value  of  the  series  is 
1  +  1  +  1  +  1+  .  .  .  .  , 
which  becomes  greater  without  limit,  the  larger  the  number 
of  terms  which  is  taken. 

Similarly,  if  x  have  any  value  greater  than  1  in  series  (2), 
the  series  has  no  limit  to  its  value. 

Hence,  in  one  kind  of  infinite  series  the  sum  approaches  a  fixed  quantity 
as  limit,  while  in  the  other  kind  the  sum  exceeds  any  assigned  limit. 

A  Convergent  Series  is  one  in  which  the  sum  of  the  first  n 
terms  approaches  a  certain  fixed  limit,  no  matter  how  great  n 
may  be. 


UNDETERMINED  COEFFICIENTS.  345 

A  Divergent  Series  is  one  in  which  the  sum  of  the  first  n 
terms  can  be  made  to  exceed  any  assigned  quantity  by  mak- 
ing n  sufficiently  large. 

332.  Use  of  Infinite  Series.  Infinite  series  are  useful  in 
representing  complex  algebraic  expressions,  since  such  series 
ordinarily  consist  of  simple  combinations  of  the  functions, 
sum,  difference,  product,  and  power  of  quantities  used.  It  is 
evident,  however,  that  infinite  series  can  be  used  in  a  valid 
way  only  when  the  series  used  is  convergent. 


Identities.  An  identity  is  a  statement  of  equality 
between  two  algebraic  expressions,  which  is  satisfied  by  all 
values  whatever  of  the  unknown  quantity  or  quantities  used 
in  the  expressions. 

Ex.  a;'-4=-(a;-2)(a;  +  2) 

is  an  identity,  since  it  is  satisfied  by  any  value  of  a;,  as  1,  2, 
3,-1,  etc.;  whereas,  for  example,  in 

7a;-6  =  10-a;, 

X  has  but  a  single  value  ;  viz.,  x  =  % 

The  correct  sign  for  an  identity  is  — ,  but  in  the  case  of 
such  identities  as  occur  in  the  early  part  of  the  algebra,  the 
custom  prevails  of  using  the  equality  mark  for  these  identi- 
ties as  well  as  for  equations.  The  student,  however,  should 
form  the  habit  of  carefully  discriminating  between  identities 
and  equalities  when  the  equality  mark  is  thus  used  for 
both. 

In  this  chapter  the  identity  mark  will  be  used  for  identities, 
and  the  equality  mark  for  equalities. 

334.  Fundamental  Property  of  Identities.  In  the  identity 
A  +  Bx-\-Cx'  +  X>ir'  +....=  A' +  B'oc  +  C  V  -f-  D'x^  +  . . . . 
(each  member  being  either  finite  or  convergent)^  the  coefficients  of 
like  powers  of  ac  are  equal;  that  is,  A=A',  B=B'f  C='C'j 
etc,  « 


346  ALGEBRA. 

For,  since  the  identity 

A  +  Bx  +  Cx'-^Dx'  +  ... .  =  J.'  +  ^'x  +  CV  4-i)V-t-  ....(1) 
is  true  for  all  values  of  x,  it  is  true  when  x  —  0. 

Hence,  from  (1)  A  =  A' (2) 

Subtracting  (2)  from  (1),  and  dividing  the  result  by  x, 

B  +  Cx-\-Dx'  +  ....=B  +  C'x+D'x'+    .    .   .    .  (3) 
This  identity  is  true  for  all  values  of  x.  , 

.  • .  letting  a;  =  0, 

B  =  B' (4) 

Subtracting  (4)  from  (3),  and  proceeding  as  before, 
C=C\    D  =  iy,  etc.,  etc. 

Ex.  Find  the  condition  that  x^  +  px  +  q  is  divisible  by  cc  +  a. 

li  x'  -\-  px -\- q  is  divisible  by  a;  +  a,  the  quotient  must  be  of 
the  form  x-\-  k. 

Hence,  x'^  +  px  +  q  =  (x  +  a)  (x  -\-  k), 

x^  +  px  -\-  q^x^  +  {a  -\-  k)x  -\-  ak. 
Hence,  by  the  fundamental  properties  of  identities, 

p  =  a  +  k (1) 

q  =  ak (2) 

Eliminating  k  from  these  equalities, 
q  =  a(p-  a), 
which  expresses  the  required  condition. 

EXERCISE  123. 

Find  the  value  of  k  that  will  make — 

1.  8a:^  4-  kx  —  91  exactly  divisible  by  2x  +  7. 

2.  6x'  +  x'^  -\-  kx  +  4:  exactly  divisible  by  3a;  —  4. 

3.  3a;'  +  kx""  +  25  exactly  divisible  by  a;'  -  3a;  +  5. 

4.  2x*  -  ar*  -  3a;'  +  lOx  +  k  divisible  by  x'  +  a;  -  2. 


.  UNDETERMINED  COEFFICIENTS.  347 

Determine  the  conditions  necessary  that— 

5.  x'^-{-ax  —  h  may  be  exactly  divisible  by  x  —  c. 

6.  ax^  +  62:  +  c  may  be  exactly  divisible  by  a;  +  d 

7.  x^  +  ax  +  h  may  be  exactly  divisible  hy  x^  —  x-\-d. 

8.  a;*  +  ax^  +  bx~  -\-  ex  ^-  d  may  be  divisible  by  x'  +  mx  -f-  n. 

Prove  that  if — 

9.  ax^  +  hx'  ^-cx  +  d  is  divisible  by  x^  —  P,  then  ad  =  be. 
10.  ax'  +  2hxy  +  6?/'  +  25ra:  -\- 2fy  +  c  is  the  square  of  a'x  +  h'y 

4-  c',  then  af  =  gh,  bg  =fh,  and  eh  =fg. 

APPLICATIONS. 
I.  Expansion  of  a  Fraction  into  a  Series. 

335.  General  Method  by  Undetermined  Coeflacients.  A 
fraction  may  be  expanded  into  a  series  by  dividing  the  nu- 
merator by  the  denominator,  but  it  is  often  more  convenient 
to  make  the  transformation  by  the  use  of  undetermined 
coefficients. 

1  — 2x' 
Ex.  Expand  — —  into  a  series  by  the  use  of  unde- 

^         l  +  2x-3x^  ^ 

termined  coefficients. 

Let  — 1  -  2x-^       ^^  +  Bx^  Cx''  +  Dx^+ (1) 

where  A,  B,  C,  D  .  .  .  .  arc  unknown  numbers  to  be  determined. 

C  lear  (1 )  of  fractions,  collecting  the  coefficients  of  x,  x*,  etc.  by  use  of 
the  vinculum  in  the  vertical  position  instead  of  by  use  of  parenthesis,  thus, 

for  {B  +  2A)x  use  +    ^1  ^• 

1  -  2x2 £EE^  +    5|  X  +        C  a:^  +       D  x»  +  .    .    .   .  (2) 
+  241         +2B  +  2C7 

-ZA  -  3P 

Since  (2)  is  an  identity,  the  coeflScients  ot  like  powers  of  x  are 
equal. 


348 

Hence, 


B+2A  =  0, 


1>      1 

j/x>.£i..a.. 

r^  =  i, 

0, 

-2, 

whence  - 

B=-2, 
0  =  5, 

0,       J 

i)  =  -  16, 
etc. 

D  +  2C-SB  =  0, 
etc. 
Substitute  for  ^,  -B,  C,  i),  .  .  .  .  in  (1), 

— L=_2£i! —  =  1  -2x  +  5x^  -  IQx^  + 
1  +  2a;  -  Sx'' 

This  result  may  also  be  obtained  by  division. 


,  Series. 


336.  Special  Cases.  When  the  lowest  power  of  x  in  the 
denominator  is  higher  than  the  lowest  power  in  the  numer- 
ator, we  may  proceed  in  either  of  two  ways,  as  illustrated  by 
the  following  example: 

1  +  2x 
Ex.  Expand  — — — -.  into  a  series. 

1  +  2a;  1  /I  +  2a;> 


Let 


1  +  2a; 
1  -3a; 
Solving  this  identity, 
1  +  2a; 


1  /I  +  2a;\ 
x'\l-  Sx) 


Sx 


A  +  Bx  +  Cx^  +  Dx^  + 


=  1  +  5a;  +  15a;2  +  45a;'  + 


Multiplying  this  result  by  — ->  that  is,  by  a;"'^,  we  obtain 


1  +  2a; 
a;2-3a;3 


=  a;-2  +  5a;-i  +  15  +  45a;  +  . 


The  required  series. 


Or,  at  the  outset  we  might  have  divided  the  first  term  of  the  numerator, 
1,  by  x^,  the  first  term  of  the  denominator,  and  determined  a;~^  as  the 
power  of  x  occurring  first  in  the  required  series,  and  let 
1  +  2x 


Ax-^  +  Bx-^  +  C+  Dx  + 
as  usual. 


x^  -  3ar» 
and  determined  A,  B,  O,  D, 

Again,  if  the  numerator  and  denominator  of  the  given  frac- 


UNDETERMINED  COEFFICIENTS.  349 

tion  contain  only  even  powers  of  x,  the  process  of  expansion 
may  be  shortened  by  using  only  even  powers  in  the  assumed 
series. 

Thus,  let  ^  _  ^^^  ^''^^^  =A-^Bx'-\-W  +  Dz'  +  ..... 


EXERCISE  124. 

Expand  into  a  series  in  ascending  powers  of  x- 


1. 

l  +  2a; 
l  +  x 

4. 

3-a;' 

l  +  2x' 

^    2-a;-4x' 
2-i-x-7? 

2. 

2-3a; 
l-2a; 

6. 

1-x-x' 

1+x^x^ 

^      Z  +  X-27? 

3. 

\-bx 

\^x-x' 

6. 

x  +  Zx' 
l-2x-\-^x' 

9.        '-^ 
2  +  6a;-a;» 

10.  2-^^-^ 2^' 

3  +  x-2x' 

1 

11. 

3-a:'  +  4x» 

2  +  x^-a;» 

12. 

l-2a: 

14. 

4  +  7a; 

l-2x'-cr:» 

X'  +  T? 

2x  +  Zx^ 

13. 

2-x' 

15. 

2  +  a;  — 4x' 

17.   1-^  +  ^'. 

x'  +  x»  2ic»  +  X*  x'  +  X* 

n.  Expansion  of  a  Radical  into  a  Series. 
337.  Illustrative  Example. 

Ex.  Expand  Vl  +  2x  into  a  series  by  use  of  undetermined 
coefficients. 

Let  VI  +  2x  =  A  +  Bx  +  Cx^  +  Dx^  + 

Squaring  both  sides,  using  method  of  Art.  89  for  the  right  member^ 

1  +  2x  =  ^'^  +  2^51  X  +      ^'1  x""  +  2AD\  x^+ 

2.401       +  2BC\ 

In  this  identity  e(juate  the  coefficients  of  like  powers  of  JC, ^.  ^^ 


350  ALGEBRA. 


1  -^  2x  El  -^  X  -  Ix^  +  ^x^  +   .  .  .  .  ,  Series. 


Another  series  may  also  be  obtained  by  taking  the  negative  value  of  A 
obtained  from  A"^  =  1, — viz.  A  =  —  1, — and  determining  corresponding  val- 
ues for  B,  C,  D,  etc. 


EXERCISE 

Expand  into  a  series  in  ascending 

125. 

powers 

of 
5. 
6. 

X — 

1.  VI  -  2x. 

3.  V4:-'Sx- 

^x\ 

+  x. 

2.  Vl  +  Ax  +  2x\ 

4.  Vd'  —  x\ 

-Zx-e^ 

m.  Separating  a  Fraction  into  Partial  Fractions. 

338.  General  Method.  If  the  degree  of  the  numerator  of 
a  fraction  be  greater  than  the  degree  of  the  denominator,  the 
fraction  may  be  converted  into  a  mixed  number,  the  fraction 
obtained  having  the  degree  of  the  numerator  less  than  the 
degree  of  the  denominator. 

If  we  consider  proper  fractions  only,  such  fractions,  if  their 
denominators  can  be  factored,  may  be  separated  into  partial 
fractions  by  the  use  of  the  properties  of  identities. 

It  is  evident  that  if  in  the  original  fraction  the  degree  of 
the  numerator  is  less  than  the  degree  of  the  denominator,  the 
same  must  be  true  in  each  partial  or  component  fraction. 

The  problem  before  us  is  the  inverse  of  that  treated  in  Arts.  143  and  144. 
There,  the  several  fractions  were  given  to  find  their  sum,  but  here,  the  sum 
is  given  to  determine  the  constituent  fractions.  Tlie  importance  and  use- 
fulness of  the  methods  here  presented  will  be  more  fully  appreciated  by  the 
pupil  whea  be  hag  advanced  to  the  Integral  Calculus. 


UNDETERMINED  COEFFICIENTS,  361 

CASE  I. 

Factors  of  the  Denominator  are  of  the  First  De- 
firree  and  Unequal. 

bx  — 14 
Ex.  1.  Separate  — — into  partial  fractions. 

a;^  —  6a;  +  8 

Let  5a: -14     ^_^4_  ^  _B_ 

a:2_6a;+8a;-2a;-4  ^  ^ 

Clearing  of  fractions, 

5a;  -  14  =  ^(a;  -  4)  +  ^(a;  -  2) (2) 

Hence,  5a;  -  14  =e  (^  +  B)x  -  4A  -  2B (3) 

Equating  coelEcients  of  like  powers  of  x, 

A  +  B  =  6,       1         whence    ^  =  2, 
-4/i -2E=  -14,J         and  ^  =  3. 

Substituting  for  A  and  ^  in  (1), 

5a;  -  14     _     2       ^      3     . 
a;2-6a;  +  8~a;-2a;-4' 

The  values  of  A  and  B  may' also  be  obtained  from  (2)  in  another  way, 
which  usually  involves  less  labor  in  this  case.     Thus,  since  (2)  is  an  iden- 
tity, and  therefore  true  for  any  value  of  x  whatever,  let  a;  =  2, 
then  from  (2)  10-14  =  ^(2-4). 

Hence  A  =  2. 

Again  in  (2)  let  a;  =  4,         then    ^  =  3. 

a;^  —  a;  —  3 

Ex.  2.  Separate  - — into  partial  fractions. 

a:^  —  4a; 

a;''-a;-3^4^^     B      .      C 


x^  —  4x  x       X  —  2      X  +  2 

Hence,  a;^  -  a;  -  3  =  ^(a;  -  2)  (a;  +  2)  +  Bx{x  +  2)  +  Gx{x  -  2)   .  (1) 
In  (1)  let    x=  0,  then    -  3  =  ^(-  2)  (+  2),        hence,    A  =  \, 

In  (1)  let    a;  =  2,  "  B=  -  \, 

In  (1)  let    a;=  -2,  ♦'  C  =  f . 

Hence,  ^'-^"zJ^A L_  + ? 

'     s^-4x    ~~Ax       8(a;-2)        8(a;  +  2) 


352  ALGEBRA. 

EXERCISE  126. 

Separate  into  partial  fractions — 

,      x-12                   ^    2x-h24  .    2a;'-13a;  +  9 

6.  — : --•  5.^ 


x{x-Z)  x^-O  2x^-%x 

2    — ?^_.  4  2a;'  +  7a;-l  x''-Q>x-l 

'   ^-x-2  '  a:(4x'-l)'  'ix'-x^-x 

--«*-- 2x^4- 4a; +  5  ax  — 14a*' 

8.   ~         :  I         •  11. 


a:'  +  x  —  2  cc^  —  3aa:  —  4a" 

2-llx-3x'  2x'  +  5ax'* 


ISx^' -  18a:' +  4a;  2a:'H-aa;-a' 

,^   a;' - 2x'  + 16        ^^       12(a;-l)  ^^         4a;- 14 

lo. •        14.  ;: •        15. 


8  a;»-4a;'  +  2a;  4x'^-20a;  +  23 

CASE  IT. 
340.  Factors  of  the  Denominator  are  of  the  First  Degree 
and  some  of  them  Repeated. 

In  this  case  some  factors  of  the  denominator  will  be  of  the 
form  x""  or  (ax  +  6)". 

q~.  -j 

If  we  consider  the  fraction  >  it  is  evidently  sep- 

a;'(a;  +  2)  ^      ^ 

arable  into  ; 1 »  the  condition  being  that 

0?  a;  +  2 

the  degree  of  the  numerator  in  each  partial  fraction  must  be 
one  less  than  the  degree  of  the  denominator. 
V\^       Aj^\Bx^C_A^    .^^.0 
'.:    ,  ar'  ~  ^         ^       ar* 

X        x^        of 
the  latter  being  the  more  convenient  form. 


*  First  reduce  to  a  mixed  number. 


UNDETERMINED  COEFFICIENTS.  363 


Hence,  we  write 


Sx~l    _A    .B  ^C   ^      D 


and  find  A,  B,  Q  D  as  usual. 

Again,  if  the  factor  of  the  denominator  which  is  a  power 
be  a  binomial,  it  is  similarly  separable. 

Thus,    J^^^.A^^^±0^±D 

^A      Bx'-ABx-^AB^-{C-VAB)x-¥D-\B 

^  X  (x~2y 

_A      B       ax-\-iy 

~~  X        x-2        (x-2)' 

__A  B         C'x-2C'-\-iy  +  20' 


x       ijc-2  {x-2y 

A    ,      B      ,       a       ^      DC 

H ~  "T- T-.  + 


^  X       x-2       {x-2y      (x-2)* 
where  A^  B,  C\  Bf'  are  numbers  to  be  determined. 

Ex.  Separate into  partial  Iractions. 

1  A    ^_B_^    a  . 


{X  -  If  {x-Vl)      x-1      {x-l}^      x  +  1 
Clearing  of  fractions, 

l  =  ^(a;2  -  1)  +  B{x  +  1)  +  C{x  -  1)> 
Hence,  1  =  (^  +  C)x^  +  {B  -  2C)x  -A+B  +  a 

Hence,  A  +  C  =  0,  ^  f  A  =  -  \. 

^  -  2C  =  0,    I        whence  \  B  =  ^. 
-A  +  B  +  C^l,  \  [C=\. 

•  '  {x~  ly  {x-i-l)"       4(«  -  1)       2{x-  1}'      4{x  +  1) 
23 


354  .  ALGEBRA, 

EXERCISE  127. 

Separate  into  partial  fractions — 


^'  xXx  +  1) 

^•(x+1)-* 

l-2x- 

IVar* 

"      x\5x- 

1) 

8x'  +  9x 

-35 

'  (x-iyQ. 

r  +  2)^ 

0.    ^'-' 

(x  +  2y  (a;-2)Xl-22;) 

^      3a;-20  ^    28a;^-4a; 

3a;'-4a;''  *  (2a;-l/* 

8a:* -81a: -54 


10. 


11. 


12. 


a:\2a;  +  3/ 

5(9  +  llx) 
(2a:^  +  a:--3)^* 

16a:' +  15a; -50 


(a; -2)*  xXx  +  1)  (2a: -5/ 

CASE   III. 

841.  One  or  More  Factors  of  the  Denominator  is  of 
the  Second  Degree. 

It  is  necessary  in  each  case  to  let  the  degree  of  the  numer- 
ator of  each  partial  fraction  be  one  less  than  the  degree  of 
the  denominator. 

X 

Ex.  1.  Separate -r—-- into  partial  fractions. 

(x  +  1)  (a:'  +  1)  ^ 

Let  ^  —     ^      4.  Bx+  O  . 

{x  +  l){x^  +  1)       x  +  1       x^  +  1 

Hence,  x^Ax^  +  A  +  Bx^  +  J5a:  +  Gc  +  0. 

x  =  {A  +  B)x''  +  {B  +  C)x  +  A^-G. 
Hence,  A+B  =  Q>,  ^  {  A  =  -  \. 

B+  C=\,   \        whence  \  B  =  \. 

^  +  c = 0,  J  yo  =  \. 

,    X - 1        ,       a;  +  1     ^ 

"  *  (x  + l)(a;2  + 1)  ~"2(a;  +  l)       2(a;''  +  l)* 


UNDETERMINED  COEFFICIENTS,  355 


Ex.  2.  Separate  -— — -  into  partial  fractions. 

ar*  +  1 

3      _     A      ^    Bx^  0 


a;'  +  l~a;  +  l      x^  -  x  ^-  \ 

Z  =  A{x''  -x  +  l)^  {Bx  +  C){x  +  1) 
3=(^  +  £)x2  +  {B  +  C-  A)x  +  A  +  Q 
Hence,  ^  +  J5  ==  0,  1  M  =  1, 

B+  C-  A^O,   I        whence  ^  B  =  -  I, 
^  +  C  =  3,  J  [  0  =  2. 

.         3      _     1 x-1 

**a;3  +  l~2;4-l      aj-'-aj  +  l 

The  following  are  other  possible  examples,  with  their  methods  of  sep- 
aration : 

3a;  -  2   ^A^Bt^^Cx'^  +  Dx^E 


x{x^  +  2}      a;  a;*  +.  2 

6a;  +  7        _    A     ^  Bx  +  C  ^  Dx  +  E 


{x  +  1)  (a;2  +2)'''      a:  +  1        3:^  +  2       {x^  +  2)» 
It  will  be  observed  that  the  number  of  known  letters  used  in  the  solution 
of  all  examples  in  Partial  Fractions  is  the  same  as  the  degree  of  the  De- 
nominator. 

EXERCISE  128. 

Separate  into  partial  fractions — 

.       5a; +  4  „    2x'-7a;  +  l  ,       lOx  +  2 


<x»  +  2)  a^-1  x'-^x'  +  l 

16x4-4  .    — 5-3a;  ^       x'  +  4a:' 


2ar'-a;-3  3  +  2x  -  2x' -  4a;»  -  3z* 

*   <2x^-3)'*  *        2x(x'-\-iy(x-l) 

IV.  Reversion  op  Series. 
342.  General  Method.     If,  for  example,  the  series 

2/  =  x  +  2a;»  +  3x'  +  4x*+ 

be  given,  it  is  impossible  by  direct  substitution  to  find  tho 
value  of  X  for  a  given  value  of  y ;  as,  ^. 


356 


ALGEBRA. 


If,  however,  we  convert  the  series  into  the  form 
X  =  ay  +  by^  -\-  cy^  +  dy*  +  .  .  .  .  , 

where  the  coefficients  a,  6,  c,  d  .  .  .  .  are  known,  the  value 
of  X  may  be  determined  by  direct  substitution  of  the  value 
of  y. 

Reversion  of  Series  is  the  process  of  converting  a  series  in 
which  y  (or  any  unknown)  is  given  in  terms  of  x  (another 
unknown)  into  a  series  in  which  x  is  given  in  terms  of  y. 


Ex.  Revert  the  series 

Let  X  =  Ay  +  By^  +  Cy^  +  By^  + 

In  (2)  substitute  for  y  its  value  from  (1), 

X  =  A{x  -  2a;2  +  3a;3  -  4a;*  + ) 

+  Bix"  -  4.t3  +  10a;*  + ) 

+  C(a:'-.6a;*+ ) 

+  i)(a;*  + ) 

+    etc. 


Hence, 


Ax 


-  2A  a;2  +  SA 

x^-4A 

+  B       -4B 

+  10B 

+  C 

-6C 

+  B 

cc*  + 


Equating  coeflficients  of  like  powers  of  x, 


^  =  1,1 

A  =  l, 

-2A  +  B  =  0, 
SA-4B  +  O  =  0, 

whence  - 

5  =  2, 
C  =  5, 

iA  +  10B-6C+  B^O, 

i)=14, 

etc. 

etc. 

(1) 

(2) 


Substituting  for  A,  B,C,  Bin  (2), 

X  =  y  +  2y^  +  57/3  +  14^  ^ 

In  case  a  given  series  contain  only  odd  powers  of  the  un- 
known number,  the  process  of  reverting  the  series  may  be 


UNDETERMINED  COEFFICIENTS.    '  357 

abbreviated  by  assuming  only  odd  powers  in  the  second  series 
used. 
Thus,  given 

p  =  x-Sx^  +  5x^-7x'  + (1) 

let  X  =  Ay  -\^  B^^  +  Qy^  +  DyT  + (2) 

For  if  we  assume  even  powers  also  of  y  in  the  second  series,  it  will  be 
"aund  that  all  even  powers  of  the  value  of  y  contain  even  powers  only 
oi"  X,  and  since  the  left-hand  member  of  (2)  contains  no  even  powers  of 
X,  all  the  coefficients  of  the  assumed  even  powers  equal  zero. 

EXERCISE  129. 

Revert  to  four  terms,  the  scries — 

1.  y  =  x  +  x''  +  x'  +  x* 

2^y  =  x-Sx'-^5x'-7x* 

5.  y  =  x  +  2x'  +  37^  +  ix* 

4.  y  =  x-ix^+ix^-ix' 

x^    .   x^       x* 
5.2^^, +  _  +  _  +  _ 

6.  y  =  ix-ir^  \x'-ix' 

7.  y  =  2x+^x''  +  4x'  +  bx'  .  .... 

8.  y=x-^ix'  +  ix'  +  lx' 

x^    ,    x^       x' 

^•^^^-3^-5-y 

10.  y  =  l-2x  +  Sx'~i7^ 

X"  7?  X* 

ll.,  =  l+.  +  ^^  +  J3+jj 


CHAPTER    XXVIII. 

BINOMIAL   THEOREM. 

FOR  POSITIVE   INTEGRAL  EXPONENTS. 

343.  General  Formula.  By  actual  multiplication  of  the 
first  few  powers  of  a  binomial,  x  -\-  a,  the  following  results 
were  obtained  and  stated  in  Art.  178: 

I.  The  Number  of  Terms  equals  the  exponent  of  the 
power  of  the  binomial,  plus  one. 

II.  Exponents.  The  exponent  of  x  in  the  first  term  equals 
the  index  of  the  required  power,  and  diminishes  by  1  in  each 
succeeding  term.  The  exponent  of  a  in  the  second  term  is  1, 
and  increases  by  1  in  each  succeeding  term. 

III.  CoeJOacients.  The  coefficient  of  the  first  term  is  1 ;  of 
the  second  term,  it  is  the  index  of  the  required  power. 

In  each  succeeding  term  the  coefficient  is  found  by  multi- 
plying the  coefficient  of  the  preceding  term  by  the  exponent  of  oc  in 
that  term,  and  dividing  by  the  exponent  of  a,  increased  by  1. 

IV.  Signs  of  Terms.  If  the  binomial  is  a  difference,  the 
signs  of  the  even  terms  are  minus ;  otherwise,  the  signs  of  all 
the  terms  are  plus. 

These  results  may  be  expressed  in  a  formula,  thus : 

(x  +  ay  =  x"  -f  nx**  -  'a  +  '^^'^~'^\  «  -  v 
,  n(n  -  1)  (n  -  2)  „    ,  ,  , 

This  formula  is  called  the  Binomial  Theorem. 
We  shall  now  prove  that  these  laws  hold  true  for  all  posi- 
tive integral  values  of  n. 
a58 


BINOMIAL  THEOREM.  369 

344.  Proof  when  ti  is  a  Positive  Integer.  The  laws  stated 
in  the  above  formula  have  been  shown  to  be  true  for  the  fourth 
power  by  actual  multiplication.  (See  Art.  178.)  We  shall  now 
prove  that  if  this  theorem  is  true  for  any  power,  the  nth,  it  is 
true  for  the  next  higher  power ;  viz.  the  n  +  1st. 

Take 

^         ^  1x2  1x2x3 

and  multiply  both  sides  hy  x  +  a. 

1x2  1x2x3 

x-^a       ,   x+a 

^         ^  1x2  1x2x3 

H-aj^a       +na;«-»a'    +  ^^^~^)a:»-2^3^. 

^ .       1x2 

(a:+a)"  +  i  =  a:"  + 1+ (n  +  l)a:"a+ r^-^^  +  n"!  x^-'a' 

rn(n-l)(n-2)^nfn-l)n  ^_2^3^ 

L  1x2x3     lx2j 

Or,       (a;  +  a)«  +  » =  re"  +  1  +  (n  +  l)a:"a  +  (^  +  l)^a:n  -  i^ts 

1  X  ^ 

(n-H)n(n-l)^n  -  j^s  + 
1x2x3 


This  is  the  same  as  the  original  formula,  except  that  n  +  1  is  used  instead 
of  n. 

By  actual  multiplication  the  formula  is  true  for  the  fourth  power.  Hence, 
by  the  general  result  just  proved  it  must  be  true  for  the  next  higher  power, 
the  fifth }  hence,  again,  for  the  sixth  power,  and  so  on  indefinitely.  This 
method  of  proof  is  called  Mathematical  Induction. 

345.  Proof  by  Use  of  Combinations.  Let  us  consider  the 
products  of  different  binomial  factors  like  x  +  a,  x  +  6,  x  +  c, 
etc.,  in  which  we  afterward  make  a  =  6  =  c,  etc. 


360 


ALGEBRA. 


{x  +  a){x  +  h) 
{x  +  a)  (a:  +  h)  {x  +  c) 

{x  +  a)  (x  +  6)  {X  +  c){x  +  d) 


a;'  +  (a  +  b)x  +  ab. 


+  a 

a;2  +  a6 

+  6 

+  ac 

+  c 

+  he 

x  +  a6e. 


x^  +  a 

a:3  +  aft 

x^  +  a6c 

+  6 

+  ac 

+  a6c^ 

+  c 

-fad 

+  acd 

+  d 

+  6c 
+  ed 

-f  6cd 

a;  +  dbcd. 


In  this  last  product  the  coefficient  of  a:*  is  1 ;  of  x^  it  is  the  sum  of  the 
combinations  of  a,  6,  c,  c?,  taken  one  at  a  time ;  of  x^  it  is  the  sum  of  the 
combinations  of  the  same  letters  taken  two  at  a  time  ;  of  a;,  the  same  taken 
three  at  a  time ;  the  last  term  is  the  product  of  a,  6,  c,  d,  taken  all 
together. 

If,  now,  6,  c,  d  each  be  made  equal  to  a,  the  left-hand  member  will  become 
{x  +  a)* ;  the  coefficient  of  a.-^  will  be  a  taken  as  many  times  as  there  are 
combinations  in  4  letters  taken  1  at  a  time  ;  that  is,  \  ;  the  coefficient  of  x^ 

will  be  a'  taken  4C2,  or   ^  times ;    of  a:,  will  be  a'  taken  ^C,,  or 


4x3x2 
1x2x3 


1x2 

times ;   the  last  term  is  a*. 


Hence,  {x  -^  aY  ^  x^  -\-  4ar'a  + 


4x3 


x'a^  + 


4x3x2 


xd?  +  a*. 


1x2  1x2x3 

It  will  now  be  a  useful  exercise  for  the  pupil  to  prove  that  if  the  above 
law  holds  for  the  product  of  any  number  of  different  binomial  factors, 

(a;  +  a)  (a;  +  6)  (a:  +  c)  .  .  .  .  (a;  +  k), 
it  will  hold  for  the  product  of  this  number  of  binomial  factors  increased  by 
1,  {X  +  I). 

Then,  letting  a  =  b==c=....=k{n  letters), 

The  product  of  n  binomial  factors  becomes  {x  +  a)** ; 
The  coefficient  of  a:**- ^  is  ^^la  =  na  ; 

a:«-Ms  „C,a2=^i^?— l)a»; 
1  X  ^ 


"  "  a:«  -  Ms  ^C^a?  = 

And  so  on  for  all  coefficients. 


n{n  —  1)  (n  -  2) 
1x2x3        ^ 


That  is,  the  general  theorem  holds  true  when  n  is  a  positi  7e  integer. 


BINOMIAL   THEOREM.  361 

346.  Other  Forms  of  the  Binomial  Formula.    When  a  is 

negative,  a',  a^,  etc.  are  negative ;  hence, 

(a;  -  ay  =  x^-  nx-  "  'a  +  ^^"^ll^x"  "  V 

-=^'jf^W. 0) 

If  in  the  original  formula  x  and  a  be  interchanged, 

(a  +  a;)"  -  a"  +  na^-^a;  +  ^^— ^a""  V 

1X2 

n(n-l)(n-2)  

1X2X3  ^  ^  ^ 

If  a  be  made  equal  to  1, 

IX.2  1X2X3  ^  ^ 

347.  Key  Number  and  rth  Term.  In  committing  the 
binomial  formula  to  memory,  it  is  helpful  to  observe  that 
a  certain  number  may  be  regarded  as  governing  the  for- 
mation of  each  of  its  terms.  This  number  is  one  less 
than  the  number  of  the  term.     Thus,  for  the  third  term  we 

have ^a:"~V,  in  which  there  are  two  factors  in  the 

1X2 

numerator  of  the  coefficient;  two  in  the  denominator;  the 
exponent  of  a:  is  n  —  2,  and  that  of  a  is  2.  Hence,  we  regard 
2  as  the  key  number  of  the  term. 

The  number  3  occurs  in  a  similar  way  in  the  formation 
of  the  fourth  term ;  4,  in  the  fifth  term,  and  so  on. 

For  the  rth  term  the  key  number  would  be  r  —  1. 

Hence, 

rth  term  =  -^^ -—- — r ^^    "^^V   \ 

r  — 1 


362                                          ALGEBRA. 
348.  Examples. 
Ex.  1.  Expand  (2a  H =1  • 

(2a  +  -^y  =  {2af  +  &(2a)*  (^)  +  ia(2a)3  (^)' 
Ex.  2.   Expand  /—  -  -A-V. 

(f-^i.r=(f--»r 

-(f)'-«(f)^(^-V-(lT(^-*)'--(lT(^-*) 

64         16  16  2  4 

Ex.  3.  Find  the  sixth  term  of  ( -  -      ^     V. 

\2      31/^7 
Use  the  formula  of  Art.  347. 

r  =  6,     n  =  9,     a;  =  ^>     a  =  -fa;~^^~i 
Ix2x3x4x  5\2/  \      '         "      ) 


BINOMIAL  THEOREM.  363 

Ex.  4.   Expand  (1  +  2a;  -  Sx'Y  by  use  of  the  binomial 
formula. 

(1+  2a;  -  Zx^Y  =  [(1  +  2x)  -  Zx^y 

=  (1  +  2xy  -  3(1  +  2xY  (3^2)  +  3(1  +  2x)  (3a;')'  -  (3x')» 
=  1  +  3(2a;)  +  3(2a;)2  +  {2xf  -  9a;2(l  +  4a;  +  4a;') 

+  27a:*(l  +  2a;)  -  27a;* 
=  1  +  6a;  +  3x2  -  28a;»  -  9a;*  +  54a;»  -  27a;*. 

EXERCISE  130. 

Expand — 

1.  (a  +  3)*.  5.  (a:*  -  2a:)^  9.  /x  - '  -  ^V- 

2.  (2a-x)^  6.  (xl/^  +  l)«. 

3. /l  +  ^y.  7.  (x-t+v^y. 

4.  (3x^  -  21/')*.  ^*  \2y  ^^)  '         12.  (  V'^  - 1/^^/. 


...(svi-y. 


"■(•^-Wi)'- 


16.  (a;'  — a; +  2)'. 

17.  (2-3a:  +  a;7. 

18.  (2x^  +  a;-3)*. 

19.  (a' +  2aa;  -  a;^*. 
15.  (3a "  ^  1/5 -  6 " * l>a)*.             20.  (3x^  -  2x  - 1)*, 


Find  the— 

21.  Sixth  term  of  (a  -  2a;0". 

22.  Eighth  term  of  (1  +  xVy)''. 

23.  Seventh  and  eleventh  terms  of  (x^  —  2/1^)^*. 

24.  Sixth  and  ninth  terms  of  (|a'6  —  2  Paf. 


25.  Tenth  and  twelfth  terms  of  fa: ^  +  ^—z\ 


26.  Middle  term  of  {Za'i-xVa)'\ 


l_ 


364  ALGEBRA, 

X  —  I  • 

28.  Term  containing  x^^  in  Ix' j   . 

fx        —Y^ 

29.  Term  containing  x'*  in  j  -  +  Vx^  J  . 

(2\^' 
a;* 1  . 

31.  Term  containing  x  in  lyVx  -\-  \-]  . 

FOR  FRACTIONAL  OR  NEGATIVE  EXPONENTS. 

349.  Examples.  The  binomial  formula  is  true  also  when 
the  exponent  of  the  binomial,  n,  is  a  fraction  or  a  negative 
number,  provided  the  resulting  series  is  convergent,  though 
no  simple  elementar}^  proof  in  this  case  can  be  given. 

Ex.  1.   Expand  (1  +  3x)^  to  4  terms. 
Using  formula  (3),  Art.  346, 

(1  +  3x)^  =  1  +  i{Sx)  +  Mi  zA)(3xy  +  Ki  -  1)  (I  -  2)(3a;)»  + 

1x2  1x2x3 


1+  f  a;  -  |a;2  +  ^^x^  + 


Ex.  2.   Expand  -^^ to  4  terms. 

Va-x 


_.  -¥-i-i)(-i-2)ixy ^ 

1x2x3  \a/ 

L        3a       9a'       81a=  .     J 


BINOMIAL  THEOREM.  365 

-i 


Ex.  3.   Find  6th  term  of 


_  1 
9'**     'f 

We  have  [Art.  347],    X  =  x^,     a  =  -  ^- —  ,    n  =  -  i,     r  =  6. 

o 

...  6thterm^-^-t--i— J^^(^.)-i-»  (_  S^'V 
1.2.3.4-5  ^    ^  V       3       / 


350.  Extraction  of  Roots  of  Numbers  by  Use  of  the 
Binomial  Formula.  When  a  number  whose  root  is  to  be 
extracted  approximates  an  exact  power  corresponding  to  the 
index  of  the  root,  the  required  root  may  often  be  obtained 
readily  by  use  of  the  binomial  formula. 

Ex.  Extract  the  cube  root  of  215  to  five  decimal  places. 
215  =  216  -  1  =  210(1  -  3j1^)  =  6'(1  -  ^h). 

1^215-=  6(1  -  A^)^ 

=  6[1  -h^h  -^'{^h?  -  Mi^hy- ] 

=  6(1  -  .001543  -  .000002  - ) 
=  5.99073+,  Boot 

EXERCISE  131. 

Expand  to  five  terms — 

1.  (a  +  x)-\ 

2.  (x'-l)^. 

3.  (x  +  3)*. 

4.  {l-\-2x'y. 

11.  (1^^-42/')*, 


5. 

/,-2           A\-3                          Q                 1 

(^      ^)     •             «•   (1  +  ^).' 

6. 

7. 

(x-'  +  xVy)    ■^.     10.   i/¥^^x. 

12.  (a    ^+  Vax-'y\ 

366  ALGEBRA. 


13.     ^  16 


1 
14. 


Vx'  -SVx 


I  -I        2x 

y  ^ 


15. 


1  r    /v  9^/   ^  ^ 


18     ^    ^  '' 


3-' ' i  18.    1 3- 

ya-'+sx-^  ivy    V 

Find  in  the  simplest  form— 

19.  Fifth  term  of  (1  +  x)^. 

20.  Eighth  term  of  (1  +  2a:)  ~  ^ 

21.  Tenth  term  of  (a'  —  Sl/ai/A 

22.  Fifth  term  of 3-- 

2a-Zv'x 

23.  Fifth  and  tenth  terms  of  (x  "  *  -  2  x/xf- 

24.  Sixth  and  eleventh  terms  of  (x~^  +  3x~^)* 

25.  Seventh  and  thirteenth  terms  of  (  V^  —  —  I 

I  2x1 

26.  The  term  containing  a;  ~  *°  in  [ o;^  — '  - 1^- 

(2        2  \4- 

28.  The  term  involving  a;"  in  -3 


Vx^  —  aVx 


Find  the  approximate  value  of  the  following  to  five  decimal 
places : 

29.  l/m  31.  V'm.  33.  1/94.  35.  1>'260. 

30.  1/65.  32.  V128.  34.  1^15.  36.  V'BB. 


CHAPTER    XXIX. 

CONTINUED   FRACTIONS. 

351.  A  Continued  Fraction  is  a  fraction  the  denominator 
of  which  contains  a  fraction,  the  denominator  of  that  fraction 
also  containing  a  fraction,  and  so  on,  either  for  a  finite  or  an 
infinite  number  of  minor  fractions. 

3 


Ex. 


4-f 


"^ 


Continued  fractions  are  usually  limited  to  those  in  which 
each  numerator  is  unity ;-  as, 

1 


c  + 


Continued  fractions  are  more  conveniently  written  and 
printed  in  the  form, 

_1_    J 1_ 

a+    b-\-    c+ 

352.  Integral  and  Converging  Fractions.  The  simple 
fractions  which  compose  a  continued  fraction  are  called  In- 
tegral Fractions.     Thus,  in  the  above  example,  -»   ->   -> 

a     0     c 

etc.  are  the  integral  fractions. 

The  Converging  Fractions  are  the  continued  fractions 

367 


S68  ALGEBRA. 

formed  by  taking  one,  two,  three,  etc.  integral  fractions  at  a 
time.     Thus,, in  the  above  evample, 

-  is  the  first  convergent, 
a 

is  the  second  convergent, 

"+6 


a+      ' 


is  the  third  convergent, 


"-I 


etc.,  etc. 

853.  A  Common  Fraction  made  into  a  Continued  Frac- 
tion. By  a  method  essentially  the  same  as  that  used  in 
finding  the  greatest  common  divisor  of  two  numbers,  a  com- 
mon fraction  may  be  made  into  a  continued  fraction.  For 
instance, 

19^    1    ^       1       ^       1 

43       43-         A        2  +  1 
19  19  19 

5 
1  1  1 


2  +  -1--        2  +  -1—       2  + 


4  1 

3+?  3+^  3+ 


1      -i 

The  process  is  more  conveniently  presented  thus: 
19)43(2 

38  The  quotients  2,  3,  1,  4,  are  the  de- 

5)19(3  nominators  of   the   integral  fractions 

15  composing  the  continued  fraction,  the 

4)5(1  numerators  in  each  case  being  1. 
4 

iH(4 


CONTINUED  FRACTIONS.  369 

854.  Computation  of  Converging  Fractions.  We  shall 
now  obtain  a  method  of  computing  the  values  of  the  suc- 
cessive convergents  of  a  continued  fraction. 

Consider  the  continued  fraction, 

J_    J        1_  J_      1      J 1^ 

a+    b+    c-h....+p+    q+    r  -\-    s4- 

Ist  convergent  =  -  > 
a 

2d  convergent  = 


^  _^  1        ab  +  1 
3d  convergent  =  —     ^  bo  +  1 


a  +  -i—        ^^"^  +  1)  +  a 

6  +  1 
c 

An  examination  of  the  third  convergent  shows  that  it  may  be  formed 
from  the  two  preceding  convergents.    Thus, 

Num.  of  3d  conv.  =  (num.  of '2d  conv.)  x  (3d  quot.)  +  (num.  1st  conv.) 
Denom.  of  3d  conv.  =  (denom.  of  2d  conv.)  x  (3dquot.)  +  (denom.  1st  conv  ) 
In  general, 
N.  ofrth  con.  =  [N.  of  (r  —  l)*^  con.]  [rth  quo.]  +  [N.  of  {r-2f^  con.] 
D.  of  rth  con.  =  [D.  of  {r  - 1}*^  con.]  [rth  quo.]  +  [D.  o/  (r  -  Sf'^  con.] 

We  shall  now  prove  that  if  these  laws  hold  for  the  formation  of  any  con- 
vergent from  the  i)receding  convergents,  they  will  hold  for  the  formation  of 
the  next  succeeding  convergent. 

Denote  the  convergents  corresponding  to  the  quotients  p,   q,  r,  s  by 

±—,     hL,     ±L,    ^,    and  suppose  the  rth  convergent  to  be  formed  accord- 
P'      Q'      R'     S'  ^^ 

ing  to  the  law. 

Hence  ^  =    ^  +  ^ (1) 

An  examination  of  the  general  continued  fraction  given  above  shows  that 

the  sth  convergent  is  formed  from  the  rth  by  changing  r  into  r  +  -' 

Making  this  substitution  in  (1), 
24 


370 


A  LQEBBA, 

s 

^(^%-)^^         Qrs^Qr^Ps 

S' 

Q\r-v\)^P^       Q'rs  +  Q^  +  P^s 

= 

{Qr  +  P)s+Q          Rs  +  Q 
{Q'r  +  P')s+Q'       E's  +  Q' 

Hence,  if  the  law  is  true  for  the  rth  convergent,  it  is  true 
for  the  next.  But  by  actual  reduction  the  law  holds  for  the 
formation  of  the  3d  convergent ;  hence,  by  the  general  prin- 
ciple just  proved  it  must  hold  for  the  4th  convergent ;  hence, 
for  the  5th,  and  so  on. 

Ex.  Find  a  series  of  converging  fractions  for  ^f|. 

Forming  the  given  fraction  into  a  continued  fraction,  the 
quotients  are, 

6,  5,       4,        3,        2. 

Hence,  convergents  are,    |,  -^V,    yW,    /^,    iri- 

The  first  and  second  convergents  are  readily  determined  from  the  contin- 
ued fraction.     For  the  others,  the  following  scheme  may  be  found  helpful : 

3d^  j4xl+I  =  -2L;  4,1,^   f3x_21+l==_68  .  ^^^^ 

14x31  +  6  =  130  13x130  +  31=421 

355.  Convergents  as  Successive  Approximations.  The 
first,  third,  ....  and  all  odd  convergents  are  larger  than  the 
value  of  the  entire  continued  fraction ;  the  second^  fourthj  .... 
and  all  even  convergents  are  smaller. 

For  consider  the  continued  fraction, 

o+    6+    c+    d+ 

The  first  convergent,  -  ?  is  larger  than  the  entire  continued 

fraction,  since  the  denominator,  a,  is  smaller  than  the  denom- 
inator of  the  entire  continued  fraction. 

The  second  convergent,  —  >  is  smaller  than  the  contin- 

"+6 


CONTINUED  FMACTIONS.  371 

ued  fraction,  since  the  denominator,  6,  is  too  small ;  hence, 

-  is  too  large  j  hence,  a  +  -  is  too  large  a  denominator;  hence, 

is  too  small.     And  so  on  alternately. 

The  convergents,  however,  approach  nearer  and  nearer  the 
value  of  the  original  fraction. 

N.  B.  Should  the  original  fraction  be  improper,  the  first  convergent  is 
an  integer,  and  of  course  less  than  the  value  of  the  fraction.  In  the  case 
of  mixed  numbers  and  improper  fractions,  therefore,  the  odd  convergents 
are  less  and  the  even  convergents  greater^  than  the  value  of  the  entire  con- 
tinued fraction. 

356.  Degree  of  Approximation  in  Convergents.    The  dif- 

p  Q 

.erence  between  two  successive  convergents,  —  and  -;»  can 
1  '  P  Q 

be  shown  to  be  j  and  therefore  the  difference  between 

P'Q 
either  of  these  convergents  and  the  value  of  the  entire  con- 
tinued fraction  is  less  than  -^r^' 

P  Q, 

The  difference  between  the  first  two  convergents  is 
lb  1         . 


a      ab  +  I      a[ab  +  1) 

We  shall  now  prove  that  if  this  law  holds  for  the  difference  between  any 
pair  of  convergents,  —  and   -^- »  it  will  hold  for  the  difference  between 

the  next  pair,  -^  >     —  • 

Let  the  symbol  "^  be  used  to  denote  the  difference  between. 

T-t  P^^9^=P9'-P'Q  =^L. (I) 


372  ALGEBRA. 

But  ^^Q-^m^-R^Q^ 

Hence,  if  the  law  is  true  for  the  difference  between  any  pair  of  converg- 
ents,  —  >     ^  >  it  ia  true  for  the  difference  between  the  next  pair,   ^ » 

■n 

—  •     But  by  actual  reduction  it  is  true  for  the  difference  between  the  first 
B'  ^ 

pair  of  convergents ;  hence,  by  the  general  principle  just  proved  it  is  true 
for  the  difference  between  the  second  pair ;   and  so  on  indefinitely. 

Ex.  In  the  example  of  Art.  354,  what  is  the  error  in  using 
the  third  convergent,  yVo^,  instead  of  the  value  of  the  entire 
continued  fraction,  -g^ff  ? 

The  next  convergent  after  yW  is  -^-^•,  hence,  the  error  is 

less  than  — - — -—  •     This  is  called  the  superior  limit  of 

the  error. 

EXERCISE  132. 

Express  as  continued  fractions — 

1.  fi.  2.  «.  3.  5H.  4.^. 

Find  the  fourth  convergent  in — 

5.3  + ~ 6.1  + -. 


2  + -^——  1  + 


4  +  -—-  3  + 


1  +  i  2  +  i 


7.1  +  ::^ 


2+    1+    2+    1  + 


3+    2+    1+    3  + 


CONTINUED  FRACTIONS.  373 

Express  the  following  as  continued  fractions,  and  find  the 
fifth  convergent  in  each  : 

9.  li.         11.  m-       13.  Uh        15-  tVoV.       17.  0.3029. 
10.  -\\\       12.  Iff.       14.  4H^.       16.  1.59.        18.  0.5678.      " 

Determine  the  superior  limit  of  the  error  in  taking  the 
fourth  convergent  for  the  continued  fraction  itself  in  each 
of  the  examples,  9-18  inclusive. 

357.  Surds  Expressed  as  Continued  Fractions.  A  quad- 
ratic surd  may  be  converted  into  an  infinite  continued  frac-* 
tion. 

Ex.  Convert  V^into  a  continued  fraction. 
2  is  the  greatest  integer  in  1/6. 
Hence,         1/6  =  2+  {V6  -  2) 

o_L  V6-  2       T/6  +  2  2 

=  -^  +  7—  X  — n =  2  + 


1  1/6  +  2  1/6  +  2 

=  2  + ^ (1) 

1/6  +  2 
2 

_  2  4-  1 /Since  2  is  the  greatest  integer  in 

2+  1^6" -2    (  l/g+2 

2  2 

=  2+ ^ . 

2+  T^6  -  2    ^  1/6  +  2 

2  1/6  +  2 

«2+-      1 


) 


2+-A 


1/6  +  2 


2+ 1 (2) 

2+ L_ 

4  +  -^— 
1/6  +  2 
2 


374  ALGEBRA. 

The  last  denominator  fraction  in  (2),  viz. >  is  the  same  as  the 

last  denominator  fraction  in  (1)  ;  hence,  continuing  the  process  indefinitely, 

V6-2+J-    -1-    -1-    J- 

2-t-     4+      2+      4+ 


An  infinite  continued  fraction  in  which  the  denominators 
repeat  themselves  periodically  is  called  a  Periodic  Continued 
Fraction. 

358.  A  Periodic  Continued  Fraction  Expressed  as  the 
Root  of  an  Equation.  A  periodic  continued  fraction  may 
be  expressed  as  the  root  of  an  equation.     Thus,  to  express 

J_    J^     _1 1_ 

2+    3+    24-    3+  ....  , 
let  X  denote  the  value  of  the  periodic  continued  fraction. 

1 

,' .  X  — 


Clearing  of  fractions, 

2a;'  +  7x  =  3  +  aJ 
a;2  +  3a;  =  | 

^  =  K-3+ VT5). 

The  sign  +  is  used  before  the  radical  I^IS,  since  x  can  have 
positive  value  only. 

EXERCISE  133. 

Express  each  surd  as  a  continued  fraction — 


1.  V5.              4.  yg. 

7.  1/T9.             10.  V^. 

2.  VZ.               5.  VI. 

8.  21/5.             11.  1/^. 

3.  V\Q.            6.  1/14. 

1 
9.  3 1/2.            12.  ^33* 

13.  3  +  1/23.         14. 

1/15  +  3          ^^     1/37  +  5 

CONTINUED  FRACTIONS.  375 

Express  each  continued  fraction  as  a  surd — 

le.  ^  ^  ^  ^ 

1+  3+  1+  3+ 

„.  JL  i  X  ^ 

2+  4+  2+  4+ 

,8.  -i-  i  A.  J. 

1+1+1+1+ 

19.1  +  JL  J-  -L  J_ 

2+  3+  2+  3+ 

20.3+^  X  J.  i_ 

4+  1+  4+  1+ 

21.4  +  ^  J-  _i_  X  A_ 

3+  1+  2+  3+  1+ 

Express  as  a  continued  fraction  the  positive  root  of  each 
equation — 

22.  x'  — 2a;  =  10.        23.  a;'--4a:  =  8.        24.  5a;' -7a;  =  2. 

25.  Express  each  root  of  3x'  —  8x  +  1  =  0  as  a  continued 
fraction. 

26.  The  circumference  of  any  circle  is  3.1415926  times  its 
diameter.  Required  the  series  of  fractions  converging  to  this 
ratio. 

27.  The  lunar  month  is  approximately  ^7.321661  days. 
Find  a  series  of  fractions  converging  to  this  quantity. 

28.  A  solar  year  is  5  hours,  48  minutes,  49  seconds  more 
than  365  days.  Find  a  series  of  common  fractions  approxi- 
mating nearer  and  nearer  the  ratio  of  this  excess  to  a  day. 


CHAPTER    XXX. 

LOGARITHMS. 

859.  The  Logarithm  of  a  number  is  the  exponent  of  that 
power  of  another  number  taken  as  the  base,  which  equals  the 
given  number.     Thus, 

If  10  be  the  base,  since  1000  =  10*,  log  1000  =  3  ; 

if  8  be  the  base,  since  4  =  8^,    log  4  =  f . 

if  B'=  N,  log3  N=L     This  is  read : 

log  of  N  to  the  base  B  =  l, 

360.  Source  of  Value  in  Logarithms.  The  source  of 
new  power  in  the  use  of  logarithms  may  be  illustrated  by 
the  multiplication  of  two  numbers  which  are  exact  powers 
of  10,  as  1000  and  100,  by  the  use  of  exponents.    Thus, 

Since  1000  ^lO*, 

and  100  =  10', 

1000  X  100  - 10^  =  100,000,  Product. 

In  like  manner,  if  361  =  lO'^-^"^'  +, 
and  29  =  10'*«^*"+, 

we  may  multiply  361  by  29,  by  adding  the  exponents  of  the 
powers  of  10  which  equal  these  numbers,  obtaining  10*°'^^^ "^, 
and  then  obtaining  from  a  table  -^f  logarithms  the  number 
corresponding  to  this  result,  which  will  be  the  product.  Thus, 
by  the  systematic  use  of  exponents  or  logarithms,  the  process 
of  multiplying  one  number  by  another  is  converted  into  the 
simpler  process  of  adding  two  numbers  (exponents). 

In  like  manner,  by  the  use  of  logarithms,  the  process  of 
376 


LOGARITHMS.  377 

dividing  one  number  by  another  is  converted  into  the  simpler 
process  of  subtracting  one  exponent  (or  log)  from  another;  the 
process  of  involution  is  converted  into  the  simpler  process  of 
multiplication;  and  evolution,  into  the  simpler  process  of 
division. 

In  the  systematic  use  of  these  properties  of  exponents  lies 
the  source  of  new  power  in  logarithms. 

361.  Systems  of  Logarithms.  Any  positive  number  ex- 
cept unity  may  be  made  the  base  of  a  system  of  loga- 
rithms. 

The  base  used  is  usually  denoted  by  placing  it  as  a  small 
subscript  to  the  word  log.  Thus,  the  logarithm  of  n  in  a 
system  whose  base  is  a  is  denoted  by  log^w. 

Two  principal  systems  of  logarithms  are  in  use — 

1.  The  Common  (or  Decimal)  or  Briggian  system,  in 
which  the  base  is  10,  used  in  numerical  operations. 

2.  The  Napierian  system,  in  which  the  base  is  2.7182818  -h, 
generally  used  in  algebraic  processes,  as  to  demonstrate  prop- 
erties of  expressions  by  the  use  of  logarithms. 

COMMON   SYSTEM. 

362.  Characteristic  and  Mantissa.  If  a  given  number, 
as  361,  be  not  an  exact  power  of  the  base,  its  logarithm,  as 
2.55751  +  for  361,  consists  of  two  parts,  the  whole  number, 
called  the  Characteristic,  and  the  decimal  part,  called  the 
Mantissa. 

To  obtain  a  rule  for  determining  the  characteristic  of  a 
given  number  (the  base  being  10),  we  have, 

10000  =  10*,         hence,     log  10000  =  4. 
1000  =  10^  "  log  1000  =  3. 

Hence,  any  number  between  1000  and  10000  has  a  log  be-' 
tween  3  and  4 ;  that  is,  it  consists  of  3  and  a  fraction.  There- 
fore, every  integral  number  consisting  of  4  figures  has  3  for  a 
characteristic. 


378  ALGEBRA. 

Similarly,  since  100--10^  10  =  10\  1=10^ 
every  number  between  100  and  1000,  and  therefore  containing 
3  integral  figures,  has  2  for  a  characteristic;  every  number 
between  10  and  100  (that  is,  every  number  containing  2  inte- 
gral figures)  has  1  for  a  characteristic;  and  every  number 
between  1  and  10  (that  is,  every  number  containing  1  inte- 
gral figure)  has  0  for  a  characteristic. 

Hence, 

The  characteristic  of  every  integral  or  mixed  number  is  one  less 
than  the  number  of  figures  to  the  left  of  the  decimal  point. 

363.  Characteristics  of  Decimal  Numbers. 
Since  1  =  lOO, 

.1-  — -10-s 
10 

.01=— =^=lo-^ 

100      10^ 

,001  = =-  — -  - 10-',  etc.,  etc., 

1000      10*  '        '        > 

the  logarithm  of  every  number  between  .1  and  1  (as,  for  in- 
stance, of  .3)  will  lie  between  —  1  and  1 ;  that  is,  will  be  —  1, 
plus  a  positive  fraction  ;  also,  the  logarithm  of  every  number 
between  .01  and  .1  (as  of  numbers  like  .0415)  will  lie  between 
—  2  and  —  1,  and  hence  consist  of  —  2,  plus  a  positive  frac- 
tion ;  and  so  on. 

Hence, 

The  characteristic  of  a  decimal  number  is  negative,  and  is,  nu- 
merically, one  more  than  the  number  of  zeros  between  the  decirnal 
point  and  the  first  significant  figure. 

The  characteristic  of  a  decimal  number  is  written  in  two 
principal  ways.     Thus, 

log  .0372  =  2.5705, 
the  minus  sign  being  placed  over  the  characteristic  2,  to  show 
that  it  alone  is  negative,  the  mantissa  being  positive, 


LOGARITHMS.  379 

We  may  also  add  and  subtract  10  from  the  given  log.  Thus, 
log  .0372  =  8.5705  -  10. 

In  practice,  we  use  the  following  rule  for  the  characteristics  of  decimal 
fractions : 

Subtract  the  number  of  zeros  between  the  decimal  point  and  the 
first  significant  figure  from  9,  and  annex  —  10  after  the  mantissa. 

864.  Mantissas  of  numbers  are  computed  by  methods  which 
are  beyond  the  scope  of  this  book.  After  being  computed  they 
are  arranged  in  tables,  and  when  needed  are  taken  from  the 
tables. 

The  position  of  the  decimal  point  in  a  number  affects  only 
the  characteristic,  not  the  mantissa. 

For  example,         69.72  =  ^^  ^  ^. 

Hence,  if  6972  =  10'^*^^^ 

log  ^^^  =:  log  - — ^  =  log  lO^-^^^''*  =  1.84336. 
5   10'  ^10' 

In  general,  log  6972  =  3.84336 

log  697.2  =  2.84336 

log  69.72  =  1.84336 

log  6.972  =  0.84336 

log  0.6972  =  9.84336 -10 

log  .06972  =  8.84336  - 10 

365.  Direct  Use  of  a  Table  of  Logarithms ;  that  is,  given 
a  number,  to  find  its  logarithm  from  the  table.  We  shall  here 
insert  a  small  table  of  logarithms,  that  the  student  may  learn 
enough  of  their  use  to  understand  their  algebraic  properties. 
The  thorough  use  of  logarithms  for  purposes  of  computation 
is  usually  taken  up  in  connection  with  the  study  of  Trigo- 
nometry. In  the  given  table  (see  pages  380,  381)  the  left- 
hand  column  is  a  column  of  numbers,  and  is  headed  N. 

The  mantissa  of  each  of  these  numbers  is  in  the  next  column  opposite. 
In  the  top  row  of  each  page  are  the  figures  0,  1,  2,  3,  4,  5,  6,  7,  8,  9. 


380 


ALGEBRA. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

414 

453 

492 

531 

569 

607 

645 

682 

719 

755 

12 

792 

828 

864 

899 

934 

969 

1004 

1038 

1072 

11(16 

13 

1139 

1173 

1206 

1239 

1271 

1303 

335 

367 

399 

430 

U 

4(51 

492 

523 

553 

584 

614 

644 

673 

703 

732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

279 

17 

304 

330 

355 

380 

405 

430 

455 

480 

504 

529 

18 

553 

577 

601 

625 

648 

672 

695 

718 

742 

765 

19 

788 

810 

833 

856 

878 

900 

923 

945 

967 

989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

222 

243 

263 

284 

304 

324 

345 

365 

385 

404 

22 

424 

444 

464 

483 

502 

522 

541 

560 

579 

598 

2:5 

617 

636 

655 

674 

692 

711 

729 

747 

766 

784 

24 

802 

820 

838 

856 

874 

892 

909 

927 

945 

962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

183 

200 

216 

232 

249 

265 

281 

298 

27 

314 

330 

346 

362 

378 

393 

409 

425 

440 

456 

28 

472 

487 

502 

518 

533 

548 

564 

579 

594 

609 

29 

624 

639 

654 

669 

683 

698 

713 

728 

742 

757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

914 

928 

942 

955 

969 

983 

997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

145 

159 

172 

33 

185 

198 

211 

224 

237 

-  250 

263 

276 

289 

302 

U 

315 

328 

340 

353 

366 

378 

391 

403 

416 

428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

6514 

5527 

5539 

5551 

36 

503 

575 

587 

599 

611 

623 

635 

647 

658 

670 

37 

682 

694 

705 

717 

729 

740 

752 

763 

775 

786 

38 

798 

809 

821 

832 

843 

855 

866 

877 

888 

899 

39 

911 

922 

933 

944 

955 

966 

977 

988 

999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

128 

138 

149 

160 

170 

180 

191 

201 

212 

222 

42 

232 

243 

253 

263 

274 

284 

294 

304 

314 

325 

43 

335 

345 

355 

365 

375 

385 

395 

405 

415 

425 

44 

435 

444 

454 

464 

474 

484 

493 

503 

513 

522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

G28 

637 

646 

656 

665 

675 

684 

693 

702 

712 

47 

721 

730 

739 

749 

758 

767 

776 

785 

794 

803 

48 

812 

821 

830 

839 

848 

857 

866 

875 

884 

893 

49 

902 

911 

920 

928 

937 

946 

955 

964 

972 

981 

60 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059!  7067 

61 

7076 

7084 

093 

101 

110 

118 

126 

135 

143i  152 

52 

160 

168 

177 

185 

193 

202 

210 

218 

226  235 

63 

243 

251 

259 

267 

275 

284 

292 

300 

308   316 

64 

324 

332 

340 

348 

356 

364 

372 

380 

388 

396 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITHMS. 


38] 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

740-^ 

7412  7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

5<i 

482 

490   497 

505 

513 

520 

528 

536 

543 

551 

57 

559 

566  574 

582 

589 

597 

604 

612 

619 

627 

58 

634 

642 

5  649 

657 

664 

672 

679 

686 

694 

701 

59 

709 

71t 

723 

731 

738 

745 

752 

760 

767 

774 

60 

7782 

7789'  7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

853 

860  868 

875 

882 

889 

896 

903 

910 

917 

62 

924 

931,  938 

945 

952 

959 

966 

973 

980 

987 

63 

993 

8000  8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

069  075 

082 

089 

096 

102 

109 

116 

122 

65 

8129 

8136  8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

195 

202  209 

215 

222 

228 

235 

241 

248 

254 

67 

261 

267 1  274 

280 

287 

293 

299 

306 

312 

319 

68 

325 

331 i  338 

344 

351 

357 

363 

370 

376 

382 

69 

388 

395 

401 

407 

414 

420 

426 

432 

439 

445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

513 

619 

525 

531 

537 

543 

549 

555 

561 

567 

72 

573 

579i  585 

591 

597 

603 

609 

615 

621 

627 

73 

633 

639;  645 

651 

657 

>663 

669 

675 

681 

686 

74 

692 

693 

704 

710 

716 

722 

727 

733 

739 

745 

75 

8751 

8756 

8762 

8768 

'8774 

8779 

8785 

8791 

8797 

8802 

76 

808 

814 

820 

825 

831 

837 

842 

848 

854 

859 

77 

865 

871 

876 

882 

887 

893 

899 

904 

910 

915 

78 

921 

927 

932 

938 

943 

949 

954 

960 

965 

971 

79 

976 

982 

987 

993 

998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

085 

090 

096 

101 

106 

112 

117 

122 

128 

133 

82 

138 

143 

149 

154 

159 

165 

170 

175 

180 

186 

83 

191 

196 

201 

206 

212 

217 

222 

227 

232 

238 

84 

243 

248 

253 

258 

263 

269 

274 

279 

284 

289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

345 

350 

355 

360 

365 

370 

375 

380 

385 

390 

87 

395 

400 

405 

410 

415 

420 

425 

430 

435 

440 

88 

445 

450 

455 

460 

465 

469 

474 

479 

484 

489 

89 

494 

499 

504 

509 

513 

518 

523 

528 

533 

533 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

590 

595 

600 

605 

609 

614 

619 

624 

628 

633 

92 

638 

643 

647 

652 

657 

661 

666 

671 

675 

680 

93 

685 

689 

694 

699 

703 

708 

713 

717 

722 

727 

94 

731 

736 

741 

745 

750 

754 

759 

763 

768 

773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9800 

9814 

9818 

96 

823 

827 

832 

886 

811 

845 

850   854 

859 

863 

97 

868 

872 

877 

881 

886 

890 

894  899 

903 

908 

98 

912 

917 

921 

926 

930 

934 

939  943 

948 

952 

99 

956 

961 

965 

969 

974 

978 

983   987 

991 

996 

N. 

0  1 

1    2 

3 

4 

5 

6 

7    8 

9 

382  ALGEBRA. 

To  obtain  the  mantissa  for  a  number  of  three  figures,  as  364,  we 
take  36  in  the  first  column,  and  look  along  the  row  beginning  with  36 
till  we  come  to  the  column  headed  4.     The  mantissa  thus  obtained  is 
.5611.     If  the  number  whose  mantissa  is  sought  contains  four  or  five 
figures,  obtain  from  the  table  the  mantissa  for  the  first  three  figures,  and 
also  that  for  the  next  higher  number,  and  subtract.     Multiply  the  differ- 
ence bettveen  the  two  mantissas  by  the  fourth  {or  fourth  and  fifth)  figure 
expressed  as  a  decimal,  and  add  the  result  to  the  mantissa  for  the  first 
three  figures .     Thus,  to  find  the  mantissa  for  167.49 
Mantissa  for  168  =  .2253 
Mantissa  for  167  =.2227 
Difference  =.0026 
Since  an  increase  of  1  in  the  number  (from  167  to  168)  makes  an 
increase  of    .0026   in  the   mantissa,    an  increase  of  .49  of  1   in  the 
number  will  make  an  increase  of  .49  of  .0026  in  the  mantissa.     But 
.0026  X  .49  =  .001274  or  .0013— 

Hence  .2227 

13 

Mantissa  for  167.49=  .2240 
Hence,  to  obtain  the  logarithm  of  a  given  number, 
Determine  the  characteristic  by  Art.  362  or  363. 

Neglect  the  decimal  point,  and  obtain  from  the  table  Cpp.  380,  381)  the 
mantissa  for  the  given  figures. 

Exs.    Log.  52.6=1.7210.        Log.  .00094  =  6.9731  —  10. 
Log.  167.49  =  2.2240.     Log.  .042308  =  8.6264  —  10. 

EXERCISE   134. 

Find  the  logarithms  of  the  following  numbers: 


1. 

37. 

7. 

175. 

13. 

.0758. 

19. 

0.7788. 

25. 

.08134. 

2. 

85. 

8. 

504. 

14. 

5780. 

20. 

.04275. 

26. 

.00-^32. 

3. 

15. 

9. 

32.9. 

15. 

.00217. 

21. 

234.76. 

27. 

95032. 

4. 

6. 

10. 

4.75. 

16. 

1275. 

22. 

5.6107. 

28. 

91706. 

5. 

90. 

11. 

.08. 

17. 

63.21. 

23. 

900.78. 

29. 

32.171. 

6. 

300. 

12. 

1.02. 

18. 

3.002. 

24. 

7781.4. 

30. 

328.07. 

366.  Inverse  Use  of  a  Table  of  Logarithms ;  that  is,  given 
«  logarithm  to  find  the  number  corresponding  to  this  logarithm,  termed 
antilogarithm; 

From  the  table  find  the  figures  corresponding  to  the  mantissa  of  the 
given  logarithm; 

Use  the  characteristic  of  the  given  logarithm  to  fix  the  decimal  point 
of  the  figures  obtained. 

Ex.    Find  the  antilogarithm  of  1.5658. 

The  figures  corresponding  to  the  mantissa,  .5658,  are  368. 

Since  the  characteristic  is  1,  there  are  2  figures  at  the  left  of  the  decl?' 
jnal  point.  'i 

.     ^eIlce,  the  antUog.  1.5658=36.8. 


LOGARITHMS.  383 

In  case  the  given  mantissa  does  not  occur  in  the  table,  obtain  from 
the  table  the  next  lower  mantissa  with  the  corresponding  three  figures  of 
the  antilogarithm.  Subtract  the  tabular  mantissa  from  the  given  man- 
tissa. Divide  this  difference  by  the  difference  between  the  tabidar  mantissa 
and  the  next  higher  mantissa  in  the  table.  Annex  the  quotient  to  the 
three  figures  of  the  antilogarithm  obtaiyied  from  the  table. 

Ex.    Find  antilog  2.4237. 

4237  does  not  occur  in  the  table,  and  the  next  lower  mantissa  is 
.4232.     The  difference  between  .4232  and  .4249  is  .0017. 

Hence  we  have  antilog        2.4237  =  265.29. 
4232 
11)  5.00  (  .29 

For  if  a  difference  of  17  in  the  last  two  figures  of  the  mantissa  makes 
a  difference  of  1  in  the  third  figure  of  the  antilog,  a  difference  of  5  in 
the  mantissa  will  make  a  difference  of  in^  of  1  or  .29  with  respect  to 
the  third  figure  of  the  antilog. 

EXERCISE    135. 

Find  the  numbers  corresponding  to  the  following  logarithms: 


1. 

1.6335. 

7. 

0.6117. 

13. 

0.4133. 

19. 

8.7727- 

-10 

2. 

2.8865. 

8. 

9.7973- 

-10. 

14. 

1.4900. 

20. 

2.4780. 

3. 

2.3729. 

9. 

7.9047- 

-10. 

15. 

3.8500. 

21. 

0.6173. 

4. 

0.5775. 

10. 

8.6314- 

-10. 

16. 

1.8904. 

22. 

1.9030. 

5. 

3.9243. 

11. 

7.7007- 

-10. 

17. 

2.4527. 

23. 

3.3922. 

6. 

1.8476. 

12. 

6.1004- 

-10. 

18. 

9.6402—10. 

24. 

9.7071- 

-10, 

367.  Properties  of  Logarithms.    It  has  been  shown  (Arts. 
199,  200)  that— 

when  771  and  n  are  commensurable.  By  the  use  of  successive 
approximations  approaching  as  closely  as  we  please  to  limits, 
the  same  law  may  be  shown  to  hold  when  m  and  n  are  incom- 
mensurable.    It  then  follows  that — 

1.  log  o6  =  log  a  +  log  &.  8.  log    a^  =  _ploga. 

2.  log  /^j  =  log  a  -  log  6.  4.  log^a  =  -^- 


384 

ALGEBRA J 

Proof  — 

Let  a  =  lO'". 

.  * .  log  a    =m. 

b  =  10". 

.  • .  log   b    =n. 

a6  =  10"*  +  ". 

.'.  log  ab    =--m  +  n  =log  a  +  log  5  . 

•  (1) 

^  =  10-^ 
b 

.  • .  log  l-\  =m  —  n  =  log  a  —  log  b  . 

•(2) 

aP^lO^rn^ 

.  • .  log   a^  =  pm  =  p  log  a     .    .    .    , 

.(3) 

■^a  =  lo". 

.(4) 

The  same  properties  may  be  proved  in  like  manner  for  a  system  of 
logarithms  with  any  other  base  than  10. 
368.    Properties  Utilized  for  Purposes  of  Oomputation. 

I.  To  Multiply  Numbers, 

Add  their  logarithms,  and  find  the  antilogarithm  of  the  sum. 
This  will  be  the  product  of  the  numbers, 

II.  To  Divide  One  Number  by  Another, 

Subtract  the  logarithm  of  the  divisor  from  the  logarithm  of  the 
dividend,  and  obtain  the  antilogarithm  of  the  difference.  This 
will  be  the  quotient. 

III.  To  Raise  a  Number  to  a  Required  Power, 

Multiply  the  logarithm  of  the  number  by  the  index  of  the  power. 
Find  the  antilogarithm  of  the  product. 

IV.  To  Extract  a  Required  Root  of  a  Number, 

Divide  the  logarithm  of  the  number  by  the  index  of  the  required 
root.     Find  the  antilogarithm  of  the  quotient. 

Ex.  1.   Multiply  527  bv  .083  by  the  use  of  logs. 

log   527  =  2.7218 

log  .083  =  a9191  -  10 

antilog  1.6409  =  53.7+,  Product. 

Ex.  2.  Compute  the  amount  of  $1  at  6%  for  20  years,  at 
compound  interest.  ' 

Amount=  (1.06)20 


log  1.06 


.0253 
20 


antilog  0.5060  =  $3.21  +,  Amount, 


LOGARITHMS.  385 

If  the  student  will  compute  the  value  of  (1.06)'^''  by  continued  multiplica- 
tion, and  compare  the  labor  involved  with  that  in  the  above  process  by  the 
use  of  logarithms,  he  will  have  a  good  illustration  of  the  value  of  loga- 
rithms. 

Ex.  3.  Extract  approximately  the  7th  root  of  15. 

log  15  =  1.1761 
■f  log  15  =  0.1680 + 

antilog  0.1680  -f  =  1.47  +,  Root 

369.  Cologarithm.  In  operations  involving  division  it  is 
usual,  instead  of  subtracting  the  logarithm  of  the  divisor,  to 
add  its  cologarithm.  The  cologarithm  of  a  number  is  ob- 
tained by  subtracting  the  logarithm  of  the  number  from  10 
—  10.  Adding  it  gives  the  same  result  as  subtracting  the 
logarithm  itself  from  the  logarithm  of  the  dividend.  The  use 
of  the  cologarithm  saves  figures,  and  gives  a  more  compact 
and  orderly  statement  of,  the  work. 

The  cologarithm  may  be  taken  directly  from  the  table  by 
use  of  the  following  rule: 

Subtract  each  figure  of  the  given  logarithm  from  9  except  the  last 
significant  figure,  which  subtract  from  10. 

Ex.  1.   Find  colog  of  36.4. 

log  36.4  =  1.5611 
colog  36.4  =  8.4389  - 10. 

8.4  X  32.4 
Ex.  2.   Compute  by  use  of  logarithms 


21/576X3.78 
log   8.4  =  0.9243 
log  32.4  =  1.5105 
colog         2  -  9.6990  -  10 
colog  1/576  =  8.6198  -  10 
colog    3.78  =  9.4225-10 

antilog  0.1761  =  1.5,  Remit. 

25 


386  ALGEBRA. 

EXERCISE  136. 

Find  by  use  of  logarithms  the  approximate  values  of— 
1.75X1.4.  ^    0.317  12.  0.48 -^  (- 1.79). 

2.  9.8X3.5.  *   -0049*  ^^    -9.91 


3.  15.1  X  .005.  8. 


78.9  -45.7 


4.831X0.25.  21  14.A:^><ii. 

.,„  9    ^22435.  23.7 

^'~  '    -O^ll  ,,    3.51X67 

9.51  11.  4.7  X  (-  0.59).  4.7  X  9.1 ' 

17.  47.1  X  3.56  X  .0079.  47  X  9  4 

18.  9.57  X  59.7  X  .0759.  ^^  ^  ^^'^ 

523  X  249 

19.  4.77  X  (-  0.71)  -^  (0.83).  ^1.  ^^—  • 

22.  (2.3)^  25.  (0.96)\  28.  1^3:29.  31.  VUM, 

28.  (6.15)^        26.  (0.795)«.        29.  1^7:65:  32.  lM7. 

24.  (3.57)*.         27.  Vl9.  30.  1^1670.  33.  V^El. 

34.  1^27:9:  37.  (2.91)^.  40.  1^19  -^  l/iS. 

35.  1^D0429:  38.  ^^iq^  41.  1^1.47  X  3.7. 

36.  (76.5)i  39.  T/30  X  1^=5.       42.  V^13  X  l^IUX 


43.  1/.005  X  1/.0765.  48.  l/OT  X  1^40  X  vW. 

44.  2^  X  7*  49.  (-  0.1)"  X  (16.3)". 

45.  l^H  X  V^.  50.  (.00812)^  -^  (31.5)A 

46.  n  X  1^^.  51.  _  (3,12)3  ^  ^^^^74-23) ». 

47.  nxVBXvn.  52.  1^7)0047^^1^0568: 


LOGARITHMS.  387 


1^  X  VIDO  (- 0.1)^  X  (0.01)^ 


-0.3384  401.8                    57.  1^:42M: 

^^'     .08659    *  *  52.37  *                 58.  l^^U2SD5: 

,,     .l-=^mm  61          121.6X9.025 

\    7.962    *  '  48.3  X  3662  X  .0856 


2563  X. 03442  VbMbX  |/61.2 

'714.8X0.511*  ^^*  ^mM" 

63.  (2.6317)*  X  (0.71272)*. 

64.  v'i:6095"-4-  V  2.945  X  V^OTTTTT. 

370.  Algebraic  Expressions  Transformed.  By  the  use 
of  properties  of  logarithms  1,  2,  3,  4,  stated  in  Art.  367,  com- 
plex algebraic  expressions  may  often  be  put  into  simpler  and 
more  convenient  forms. 


Ex.  Log  -jf--^  =  log  V¥^^-  log  (1  +  xy 

(1  +  xy 

=  ilog(a'~a:^)-21og(l+a;) 

=  i  log  (a  +  xj  +  -J  log  (a  -  x)  -  2  log  (1  +  x). 

Various  formulas  may  thus  be  put  into  a  form  adapted  to 
numerical  computation. 

For  instance,  problems  relating  to  compound  interest,  prin- 
cipal, amount,  and  time  may  be  solved  by  the  use  of  loga- 
rithms. 

Since  a  =  p(l  H-  r)"  (see  Art.  368), 

log  a  =  log  /)  +  n  log  (1  +  r). 


388  ALGEBRA. 

Hence,  log  p  =  log  a  -  ?i  log  (1  +  r) 

_  log  a  —  logp 
log(l  +  r) 

log(l+r)  =  ^^^^-i^. 

371.  Exponential  Equations.  An  exponential  equation  is 
one  of  the  form  a'  =  b,  where  a  and  b  are  known  quantities 
and  X  is  unknown. 

An  equation  of  this  kind  can  be  solved  by  taking  the  loga- 
rithm of  each  member. 

Ex.  1.  Solve  (1.06)''  —  3  (that  is,  find  the  number  of  years 
in  which  $1  will  amount  to  $3  at  6%  compound  interest). 

a;  log  1.06 -log  3 

--r^?-^  =  €?^=  18.858+  Fear., 
log  1.06       .0253 

Ex.  2.  Given  0.3^  =  2;  find  the  value  of  x. 
Taking  the  logarithm  of  each  member  of  the  equation, 
a;  log  (0.3)  =  log  2. 

Hence,  ,  == -ML  ^  _0:301^ 

log  0.3      9.4771-10 

\^_a30io.^_ 

-0.5229 
Ex.  3.  Given  a'  =  6^*  +  ^ ;  find  the  value  of  x. 
Taking  the  logarithm  of  each  member  of  the  equation, 

a;loga  =  (2x  +  l)log6. 
Whence,  a;  log  a  —  2x  log  6  =  log  6 

log  b 
log  a  —  2  log  b 


LOGARITHMS.  389 

GENERAL   PROPERTIES   OF   SYSTEMS   OF  LOGA- 
RITHMS. 

372.  Logarithm  of  Unity.  In  any  system  of  logarithms 
the  logarithm  of  1  is  zero. 

For,  taking  a  as  the  base, 

a'  =  \.         .-.logJ.^O. 

373.  Logarithm  of  the  Base.  In  any  system  the  loga- 
rithm of  the  base  itself  is  unity. 

For     a}=^a.     .*.  loga  =  l. 

374.  Logarithm  of  Zero.  In  any  system  whose  base  is 
greater  than  unity  the  logarithm  of  zero  is  minus  infinity ; 
that  is,  as  the  number  approaches  0,  the  logarithm  ap- 
proaches minus  infinity. 

For,  if  a>  1,  a-«  =  —  -  —  =  0. 

a*       oo 

.  •  ..log„0=-  — 00. 
But  in  any  system  whose  base  is  less  than  unity,  the  loga- 
rithm of  zero  is  plus  infinity. 

For    a"^  =■-  0,     since  a  <  1. 

.  •  .   loga  0  ^  QO  .  • 

375.  Change  of  the  Base  of  a  System  of  Logarithms. 

Let  a  and  h  be  the  bases  of  two  systems  of  logarithms,  and  n 
be  the  number.    Then 

log,„  =  |2g^ (1) 

Let      a*  =  n;    hence,  a:  =  loga^, 
and     fe''  =  n;         "      y  =  \ogi,n, 

X 

Also,    a"  =  6^        ,'.  a"  =--h. 
.-.  loga6  =  -,  or  y 


y  loga6 


1  loga  n 

loga6 


390  ALQEBftA. 

Hence  the  logarithm  of  a  number,  n,  in  a  system  whose 
base  is  a,  being  given,  the  logarithm  of  the  same  number  in 
a  system  with  another  base,  6,  may  be  found  by  dividing  the 
given  logarithm  by  the  logarithm  of  b  in  the  system  whose 
base  is  a. 

From  the  above  relation  (1)  we  may  also  prove 

logft  a  X  loga  h  =  \. 
For  putting  n  =  a  in  (1), 

,  logo «  1 

loga  6  loga  6 

.-.  logft  a  X  loga  6  =  1. 
876.  Examples. 
Ex.  1.   Find  the  logarithm  of  0.7  to  the  base  5. 

ByArt.375,log5  0.7  =  ^^^^ 

^  9.8451  - 10 
0.6990 
—  0 1549 
=  06990^  ^  ~ ^'^^^^ ■^'  ^^^^''^*^^^- 

Ex.  2.  Find  the  logarithm  of  243  to  the  base  9  without  the 
use  of  the  tables. 

Let  log9  243  =  a; ;    then  9^  =  243. 

Hence,  {Zy  =  ^\    or     3'*  =  3^ 

Hence,  2a;  =  5 

a;  ==  f ,  Logarithm, 

Ex.  3.  Find  the  number  of  digits  in  5^^ 

Log  (5^«)  - 16  log  5  =  16  X  0.6990 
- 11.1840. 

Since  the  characteristic  is  11,  the  number  of  digits  in  5^^ 
is  12. 


LOGARITHMS.  391 

EXERCISE  137. 

Find  the  approximate  value  of  x  in  each  of  the  following 
exponential  equations : 

1.  40-  =  75.  6.  6^  +  ^  =  17.  11.  20  =  5. 

2.  20' =  100.  7.  3" -'•  =  5.  12.  27'""' =  8. 
3.5^  =  12.               8.  12*  +  *-45^           13.  0.7'=  =  0.3. 

4.  7^  =  25.  9.  5^-»*=:2*  +  ^         14.  13.18*  =  .0281. 

6.  1.3^  =  7.2.  10.  (0.4) -="  =  7.  15.  0.703^  =  1.09. 

Express  in  terms  of  log  a>  log  6,  log  c,  and  log  x — ^ 
16.  loga'6».  21.  log  ^'^^. 


17.  logax^. 

18.  log  6'!^.  22.  log 

0  vc 

19.  log  V'a^ '  V^,       '  23.  log  a " 'h'  Vc=^. 

20.  log  \/--  24.  log 


bcVx 


Express  the  value  of  x  in  terms  of  the  logs  of  the  known 
quantities  in  each  of  the  following  equations : 

25.  a*  =  76^  •        29.  Bo' + '  =  a'b'^  ^  \ 
2Q,  (f  =  a'-\  SO.  Q(fb^''  =  ll(a- by. 

27.  13*+^  =  a'6.  31.  a'^-ft' =  (2a- 1/'. 

28.  7a'  =  3^^  32.  25a^^  =  Va'-l}\ 

Find  the  values  of — 

33.  logsSO.  86.  log6  4.5.  39.  logs  0.9. 

34.  logs  12.  37.  logi2  8.  40.  logo.sGS. 

35.  Iog9o25.  88.  log4.t23.  41.  log^.j>/3. 


392  ALGEBRA. 

Find  without  the  use  of  the  tables  the  base  when — 

42.  log,  9  =  |.        44.  log,  5  =  0.5.  46.  log,  3f  =  -  f . 

43.  log,  8  =  |.        45.  log,  iV  =  -  0.8.        47.  log,3 1/3  -  1.5. 

Find  without  the  use  of  the  tables — 

48.  logaie.  50.  log9^.  52.  Iog2.0625. 

49.  logs  4.  51.  log^^ie.  53.  logyi4T/2. 

Find  the  number  of  digits  in — 
64.  (875)^«.  55.  T\  56.  9^.  67.  57". 

Show  that — 

68.  (|i)^«'>100.  59.  (|^y°^>  100000. 

60.  There  are  more  than  300  zeros  between  the  decimal 
point  and  the  first  significant  figure  in  (0.5)^°^. 

In  the  following  geometrical  progressions  find  n : 
61.  Given  a,  r,  I.  63.  Given  a,  Z,  s, 

^2.  Given  a,  r,  s.  64.  Given  r,  Z,  s. 

65.  Given  a  =  2,  r=:5,  Z  =  31,250. 

66.  Given  a  =  -^,  r  =  3,  s  =  364|. 

67.  Given  a  =  0.375,  Z  =  96,  s  =  191|. 

68.  Find  the  amount  of  $1000  for  20  years  at  5%  compound 
interest, 

09.  Find  the  amount  of  $875  for  18  j^ears  at  6%  compound 
interest. 

70.  How  long  will  it  require  for  a  sum  of  money  to  double 
itself  at  5%  compound  interest?    At  7  per  centum? 

71.  If  $1280  amounts  to  $37,770  in  50  years  at  compound 
interest,  what  is  the  rate  per  centum? 


rjHAPTER    XXXI. 
HISTORYi^OF  ELEMENTARY  ALGEBRA. 

377.  Epochs  JH  \e  Development  of  Algebra.  Some 
knowledge  of  thif  origin  and  development  of  the  symbols 
and  processes  of  Algebra  is  important  to  a  right  under- 
standing of  them. 

The  oldest  mathematical  treatise  known  is  a  papyrus  roll, 
now  in  the  British  Museum,  entitled  "  Directions  for  Attain- 
ing to  the  Knowledge  of  All  Dark  Things."  It  was  written 
by  a  scribe  named  Ahmes  at  least  1700  b.  c,  and  is  a  copy, 
the  writer  says,  of  a  more  ancient  work,  dating,  say,  3000  b.  c., 
or  several  centuries  before  jthe  time  of  Moses.  This  papyrus 
roll  contains,  among  other  things,  the  beginnings  of  algebra 
as  a  science.  Taking  the  epoch  indicated  by  this  work  as  the 
first,  the  principal  epochs  in  the  development  of  algebra  are 
as  follows: 

1.  Bg3^tian :  3000  B.  C.-1500  B.  O. 

2.  Greek  (at  Alexandria)  :  200  A.  D.-400  A.  D.  Princi- 
pal writer,  Diophantus. 

3.  Hindoo  (in  India)  :  500  A.  D.-1200  A.  D. 

4.  Arab  :  800  A.  D.-1200  A.  D. 

5.  European :  1200  A.  D.-.  Leonardo  of  Pisa,  an  Italian, 
published  a  work  in  1202  a.  d.  on  the  Arabic  arithmetic,  but 
containing  an  account  also  of  the  science  of  algebra  as  it 
then  existed  among  the  Arabs.  From  Italy  the  knowledge 
of  algebra  spread  to  France,  Germany,  and  England,  where 
its  subsequent  development  took  place. 

We  will  consider  briefly  the  history  of — 

293 


394  ALGEBRA. 

I.  Algebraic  Symbols. 
II.  Ideas  of  Algebraic  Quantity,  "jt 

III.  Algebraic  Processes.  It. 

\ 

I.  History  of  Algebraic  j  /mbols. 

378.  Symbol  for  the  Unknown  Qua/.atity. 

1.  Egyptians  (1700  b.  c.)  used  the  wr  ^"  hau  "  (expressed, 
of  course,  in  hieroglyphics),  meaning  •"Lvlap." 

2.  Diophantus  (Alexandria,  350  a.  d.  ?),  ?',  or  ?°  ;  plu- 
ral, 9?. 

3.  Hindoos  (500  a.  D.-1200  a.  d.),  Sanscrit  word  for  "  color," 
or  first  letters  of  words  for  colors  (as  of  "  blue,"  "  yellow," 
"  white,"  etc.). 

4.  Arabs  (800  a.  D.-1200  a.  d),  Arabic  word  for  "thing  "or 
"  root "  (the  term  "  root,"  as  still  used  in  algebra,  originates 
here). 

6.  Italians  (1500  a.  d.).  Radix,  R,  Rj. 

6.  Bombelli  (Italy,  1572  a.  d.),  vL'. 

7.  Stifel  (Germany,  1544),  A,  B,  C,  .  .  .  , 

8.  Stevinus  (Holland,  1586),  0. 

9.  Vieta  (Prance,  1591),  vowels  A,  E,  /,  0,  U. 
10.  Descartes  (France,  1637),  x,  y,  z,  etc. 

379.  Symbols  for  Powers  (of  x  at  first).     Exponents. 

1.  Diophantus,  duva/j.i<;^  or  d"  (for  sqiaare  of  the  unknown 
quantity);    xujSog^  or  x"  (for  its  cube). 

2.  Hindoos,  initial  letters  of  Sanscrit  words  for  "  square  " 
and  "cube." 

3.  Italians  (1500  a.  d.),  "census"  or  "zensus"  or  "z"  (for 
x')  ;  "  cubus  "  or  "  c  "  (for  x'). 

4.  Bombelli  (1579),  vl^,  e^,  ^  (for  x,  x\  x'). 

5.  Stevinus  (1586),  ©,  ©,  ©  (for  x,  x\  x'). 

6.  Vieta  (1591),  A,  A  quadrnius,  A  cubus  (for  x,  x\  x^), 

7.  Harriot  (England,  1631),  a,  aa,  aaa. 


HISTORY  OF  ELEMENTARY  ALGEBRA.  395 

8.  Herigone  (France,  1634),  a,  a2,  a3. 

9.  Descartes  (France,  1637),  x,  x^,  x^. 

Wallis  (England,  1659)  first  justified  the  use  of  fractional 
and  negative  exponents,  though  their  use  had  been  suggested 
before  by  Stevinus  (1586). 

Newton  (England,  1676)  first  used  a  general  exponent,  as 
in  x**,  where  n  denotes  any  exponent,  integral  or  fractional, 
positive  or  negative. 

380.  Symbols  for  EZnown  Quantities. 

1.  Diophantus,  /j^ovade?  (i.  e.  monads),  or  /j.°  . 

2.  Regiomontanus  (Germany,  1430),  letters  of  the  al- 
phabet. 

3.  Italians,  d,  from  ''  dragma." 

4.  Bombelli,  \^. 
6.  Stevinus,  ©. 

6.  Vieta,  consonants,  B,  Q,  D,  F,  .  ... 

7.  Descartes,  a,  b,  c,  d. 

Descartes  possibly  used  the  last  letters  of  the  alphabet,  x,  y,  z,  to  denote 
unknown  quantities  because  these  letters  are  less  used  and  less  familiar  than 
a,  b,  c,  d,  .  .  .  .  ,  which  he  accordingly  used  to  denote  known  numbers. 

381.  Addition  Sign.     The  following  symbols  were  used  : 

1.  Egyptians,  pair  of  legs  walking  forward  (to  the  left),  j\., 

2.  Diophantus,  juxtaposition  (thus,  ab,  meant  a  +  b). 

3.  Hindoos,  juxtaposition  (survives  in  Arabic  arithmetic, 
as  in  2f ,  which  means  2  +  f ). 

4.  Italians,  "  plus,"  then  "p  "  (or  e,  or  0). 

5.  Germans  (1489),  -+-,+,+. 

382.  Subtraction  Sign. 

1.  Egyptians,  pair  of  legs  walking  backward  (to  the  right), 
thus,  ZV.  ;  or  by  a  flight  of  arrows. 

2.  Diophantus,  ^  (Greek  letter  i^  inverted). 

3.  Hindoos,  by  a  dot  over  the  subtracted  quantity  (thus, 
mn  meant  m  —  n). 


396  ALGEBRA. 

4.  Italians,  "  minus,"  then  M  or  m,  or  de. 

6.  Germans  (1489),  horizontal  dash,  — . 

The  signs  +  and  —  were  first  printed  in  Johann  Widman's  Mercantile 
Arithmetic  (1489).  These  signs  probably  originated  in  German  warehouses, 
where  they  were  used  to  indicate  excess  or  deficiency  in  the  weight  of  bales 
and  chests  of  goods.  Stifel  (1544)  was  the  first  to  use  them  systematically 
to  indicate  the  operations  of  addition  and  subtraction. 

383.  Multiplication  Sign.  Multiplication  at  first  was  usu- 
ally expressed  in  general  language.     But— 

1.  Hindoos  indicated  multiplication  by  the  syllable  "  6/ia," 
from  "  bharita,"  meaning  *'  product,"  written  after  the  factors. 

2.  Oughtred  and  Harriot  (England,  1631)  invented  the 
present  symbol,  X. 

3.  Descartes  (1637)  used  a  dot  between  the  factors  (thus, 
a-b). 

384.  Division  Sign. 

1.  Hindoos  indicated  division  by  placing  the  divisor  under 
the  dividend  (no  line  between).     Thus,  a  meant  c^  d. 

2.  Arabs,  by  a  straight  line  (thus,  a  —  6,  or  a  I  6,  or  -7-  ]• 

3.  Italians  expressed  the  operation  in  general  language. 

4.  Oughtred,  by  a  dot  between  the  dividend  and  divisor. 

5.  Pell  (England,  1630),  -^. 

385.  Equality  Sign. 

1.  Egyptians,  ^  .  ^  i  (also  other  more  complicated 
symbols  to  indicate  different  kinds  of  equality). 

2.  Diophantus,  general  language  or  the  symbol,   K 

3.  Hindoos,  by  placing  one  side  of  an  equation  immediately 
under  the  other  side. 

4.  Italians,  se  or  a ;  that  is,  the  initial  letters  of  "  sequalis  " 
(equal).  This  symbol  was  afterward  modified  into  the  form, 
X),  and  much  used,  even  b}^  Descartes,  long  after  the  invention 
of  the  present  symbol  by  Recordje. 


HISTORY  OF  ELEMENTARY  ALGEBRA.  397 

.   6.  Recorde  (England,  1540),  =^. 

He  says  that  he  selected  this  symbol  to  denote  equality  because  "than 
two  equal  straight  lines  no  two  things  can  be  more  equal." 

386.  Other  Symbols  used  in  Elementary  Algebra. 

Inequality  Signs  (>  <)  were  invented  by  Harriot  (1631). 

Oughtred,  at  the  same  time,  proposed  |,  |  as  signs  of  inequal- 

ity, but  those  suggested  by  Harriot  were  manifestly  superior. 

Parenthesis,  (  ),  was  invented  by  Girard  (1629). 

The  Vinculum  had  been  previously  suggested  by  Vieta 
(1591). 

Radical  Sign.  The  Hindoos  used  the  initial  syllable  of 
the  word  for  square  root,  "  Ka,"  from  '•  Karania,"  to  indicate 
square  root. 

Rudolf  (Germany,  1525)  suggested  the  symbol  used  at  pres- 
sent  ( ]/)  (the  initial  letter,  r,  in  the  script  form,  of  the  word 
"  radix,"  or  root)  to  indicate  square  root,  m/  to  denote  the 
4th  root,  and  mV  to  denote  cube  root. 

Girard  (1633)  denoted  the  2d,  3d,  4th,  etc.  roots,  as  at  pres- 
ent by  1^,  y",    1^,  etc. 

The  sign  for  Infinity,  oo ,  was  invented  by  Wallis  (1649). 

387.  Many  other  Algebraic  Symbols  have  been  invented 
in  recent  times,  but  these  do  not  belong  to  elementary  algebra. 

Other  kinds  of  algebra  have  also  been  invented  employing 
other  systems  of  the  symbols. 

388.  General  Illustration  of  the  Evolution  of  Algebraic 
Symbols.  The  following  illustration  will  serve  to  show  the 
principal  steps  in  the  evolution  of  the  symbols  of  algebra : 

At  the  time  of  Diophantus  the  numbers  1,  2,  3,  4,  .  ...  were  denoted 
by  letters  of  the  Greek  alphabet,  with  a  dash  over  the  letters  used;  as, 
a,  "^,  7,   .  .  .  . 

In  the  algebra  of  Diophantus  the  coefficient  occupies  the  last  place  in  a 
term  instead  of  the  first  as  at  present. 


398  ALGEBRA. 

Beginning  with  Diophantus,  the  algebraic  expression, 
x'  +  5x  —  4,  would  be  expressed  in  symbols  as  follows : 

3" a  ?6  j/fx  ix°  d  (Diophantus,  350  a.  d.). 

Iz  p.5  RmA  (Italy,  1500  a.  d.). 

lQ  +  bN-4:  (Germany,  1575). 

lv!y  p.5v^  mA^  (Bombelli,  1579). 

1©  +5©  -4®  (Stevinus,  1586). 

lAq  +  bA  -4  (Vieta,4591). 

latt+5a-4  (Harriot,  1631). 

Ia2  +  5al  -  4  (Herigone,  1634). 

7?  +  bx    —  4  (Descartes,  1637). 

889.  Three  Stages  in  the  Development  of  Algebraic 
Symbols. 

1.  Algebra  without  Symbols  (called  Rhetorical  Alge- 
bra). In  this  primitive  stage  algebraic  quantities  and 
operations  are  expressed  altogether  in  words,  without  the 
use  of  symbols.  The  Egyptian  algebra  and  the  earliest 
Hindoo,  Arabian,  and  Italian  algebras  are  of  this  sort. 

2.  Algebra  in  which  the  Symbols  are  Abbreviated 
Words  (called  Syncopated  Algebra).  For  instance,  "j9" 
is  used  for  "  plus." 

The  algebra  of  Diophantus  is  mainly  of  this  sort.  Euro- 
pean algebra  did  not  get  beyond  this  stage  till  about  1600  a.  d. 

3.  Symbolic  Algebra.  In  its  final  or  completed  state  al- 
gebra has  a  system  of  notation  or  symbols  of  its  own,  inde- 
pendent of  ordinary  language.  Its  operations  are  performed 
according  to  certain  laws  or  rules,  "  independent  of,  and  dis- 
tinct from,  the  laws  of  grammatical  construction." 

Thus,  to  express  addition  in  the  three  stages  we  have 
"plus,"  p,  +  ;  to  express  subtraction,  "minus,"  w,  — ;  to  ex- 
press equality,  "  sequalis,"  ^,  ^. 

Along  with  the  development  of  algebraic  symbolism  there 


HISTORY  OF  ELEMENTARY  ALGEBRA.  399 

was  a  corresponding  development  of  ideas  of  algebraic  quan- 
tity and  of  algebraic  processes. 

n.  History  op  Algebraic  Quantity. 

390.  The  Kinds  of  Quantity  considered  in  algebra  are 
positive  and  negative ;  particular  (or  numerical)  and  gen- 
eral; integral  and  fractional;  rational  and  irrational; 
comHiensurable  and  incommensurable;  constant  and 
variable;    real   and   imaginary. 

391.  Ahmes  (1700  b.  c.)  in  his  treatise  uses  particular,  pos- 
itive quantity,  both  integral  and  fractional  (his  fractions,  how- 
ever, are  usually  limited  to  those  which  have  unity  for  a 
numerator).  That  is,  his  algebra  treats  of  quantities  like  8 
and  I",  but  not  like  —3,  or  —  f,  or  V2,  or  —a. 

392.  Diophantus  (350  a.  d.)  used  negative  quantity,  but  only 
in  a  limited  way ;  that  is,  in  connection  with  a  larger  positive 
quantity.  Thus,  he  used  7—5,  but  not  5  —  7,  or  —  2.  He 
did  not  use,  nor  apparently  conceive  of,  negative  quantity 
having  an  independent  existence. 

393.  The  Hindoos  (500  a.  D.-1200  a.  d.)  had  a  distinct 
idea  of  independent  or  absolute  negative  quantity,  and  used  the 
minus  sign  both  as  a  quality^sign  and  a  sign  of  operation. 
They  explained  independent  negative  quantity  much  as  is 
done  to-day  by  the  illustration  of  debts  as  compared  with 
assets,  and  by  the  opposition  in  direction  of  two  lines. 

Pythagoras  (Greece,  520  b.  c.)  discovered  irrational  quantity, 
but  the  Hindoos  were  the  first  to  use  this  in  algebra. 

394.  The  Arabs  avoided  the  use  of  negative  quantity  as  far 
as  possible.  This  led  them  to  make  much  use  of  the  process 
of  transposition  in  order  to  get  rid  of  negative  terms  in  an 
equation.  Their  name  for  algebra  was  "  al  gebr  we'l  mukabala," 
which  means  transposition  and  reduction. 

The  Arabs  used  surd  quantities  freely, 


400  ALGEBRA. 

395.  In  Europe  the  free  use  of  absolute  negative  quantity 
was  restored. 

Vieta  (1591)  was  principally  instrumental  in  bringing  into 
use  general  algebraic  quantity  (known  quantities  denoted  by 
letters  and  not  figures). 

Cardan  (Italy,  1545)  first  discussed  imaginary  quantities, 
which  he  termed  "  sophistic  "  quantities. 

Euler  (Germany,  1707-83)  and  Gauss  (Germany,  1777- 
1855)  first  put  the  use  of  these  quantities  on  a  scientific  basis. 

Descartes  (1637)  introduced  the  systematic  use  of  variable 
quantity  as  distinguished  from  constant  quantity. 

ni.  History  of  Algebraic  Processes. 

396.  Solution  of  Equations.  Ahmes  solves  many  simple 
equations  of  the  first  degree^  of  which  the  following  is  an  example: 

"  Heap  its  seventh,  its  whole  equals  nineteen.    Find  heap." 
In  modern  S3"mbols  this  is, 

Given  -  +  a;  =^  19 ;  find  x. 

7 

The  correct  answer,  16f ,  is  given  by  Ahmes. 
Hero  (Alexandria,  120  b.  c.)  solved  what  is  in  effect  the 
quadratic  equation^ 

\\d'  +  ''-^d  =  s, 

where  d  is  unknown,  and  s  is  known. 

Diophantus  solved  simple  equations  of  one,  and  simulta- 
neous equations  of  two  and  three  unknown  quantities.  He 
solved  quadratic  equations  much  as  is  done  at  present,  com- 
pleting the  square  by  the  method  given,  in  Art.  255.  How- 
ever, in  order  to  avoid  the  use  of  negative  quantity  as  far  as 
possible,  he  made  three  classes  of  quadratic  equations,  thus, 


ax' 

+  bx  =  c, 

ax' 

-\-c   =6x, 

ax' 

=  6a;  +  c. 

HISTORY  OF  ELEMENTARY  ALGEBRA.  401 

In  solving  quadratic  equations  he  rejected  negative  and 
irrational  answers. 

He  also  solved  equations  of  the  form  ax""  —  6a;^ 

He  was  the  first  to  investigate  indeterminate  equations,  and 
solved  many  such  equations  of  the  first  degree  with  two  or 
three  unknown  quantities,  and  some  of  the  second  degree. 

The  Hindoos  first  invented  a  general  method  of  solving  a 
quadratic  equation  (now  known  as  the  Hindoo  method,  see 
Art.  256).  They  also  solved  particular  cases  of  higher  de- 
grees, and  gave  a  general  method  of  solving  indeterminate 
equations  of  the  first  degree. 

The  Arabs  took  a  step  backward,  for,  in  order  to  avoid  the 
use  of  negative  terms,  they  made  six  cases  of  quadratic  equa- 
tions; viz.: 

ax^  =  bxj  ax"^  +  6a;  =  c, 

ax^  =  Cj  ax^  -{-  c   =  6'r, 

bx  =c,  ,  ax^  =  hx-\-  c. 

Accordingly,  they  had  no  general  method  of  solving  a  quad- 
ratic equation. 

The  Arabs  also  solved  equations  of  the  form  ax^^  +  hx^  =  c, 
and  obtained  a  geometrical  solution  of  cubic  equations  of  the 
form  x"  +  pa;  +  g  =  0. 

In  Italy,  Tartaglia  (1500-59)  discovered  the  general  solution 
of  the  cubic  equation,  now  known  as  Cardan's  solution.  Ferrari, 
a  pupil  of  Cardan,  discovered  the  solution  of  equations  of  the 
fourth  degree. 

Vieta  discovered  many  of  the  elementary  properties  of  an 
equation  of  any  degree;  as,  for  instance,  that  the  number  of 
the  roots  of  an  equation  equals  the  degree  of  the  equation. 

397.  Other  Processes.  Methods  for  the  Addition,  Sub- 
traction, and  Multiplication  of  polynomial  expressions  were 
given  by  Diophantus. 

Transposition  was  first  used  by  Diophantus,  though,  as  a 
process;  it  was  first  brought  into  prominence  by  the  Arabs. 


402  ALGEBRA. 

The  Square  and  Cube  Root  of  polynomial  expressions 
■were  extracted  by  the  Hindoos. 

The  methods  for  using  Radicals,  including  the  extraction 
of  the  square  root  of  binomial  surds  and  rationalizing  the 
denominators  of  fractions,  were  also  invented  by  the  Hindoos. 

The  methods  of  using  fractional  and  negative  Exponents 
were  determined  by  Wallis  (1659)  and  Sir  Isaac  Newton. 

The  three  Progressions  were  first  used  by  Pythagoras  569 
B.  C.-500  B.  c.). 

Permutations  and  Combinations  were  investigated  by 
Pascal  and  Fermat  (France,  1654). 

The  use  of  Undetermined  CoeflQcients  was  introduced  by 
Descartes. 

The  Binomial  Theorem  was  discovered  by  Newton  (1655), 
and,  as  one  of  the  most  notable  of  his  many  discoveries,  is 
said  to  have  been  engraved  on  liis  monument  in  Westminster 
Abbey. 

Continued  Fractions  were  first  used  by  Cataldi  (Italy, 
1653),  though  none  of  their  properties  were  demonstrated 
by  him.  Lord  Brouncker  (England,  1620-84)  was  the  first  to 
do  this. 

Logarithms  were  invented  by  Lord  Napier  (Scotland,  1614) 
after  a  laborious  search  for  means  to  diminish  the  work  in- 
volved in  numerical  computations,  and  were  improved  by 
Briggs  (England,  1617). 

The  fundamental  Laws  of  Algebra  (the  Associative,  Com- 
mutative, and  Distributive  Laws ;  see  Arts.  33-36)  were  first 
clearly  formulated  by  Peacock  and  Gregory  (England,  1830- 
45),  though,  of  course,  the  existence  of  these  laws  had  been 
implicitly  assumed  from  the  beginnings  of  the  science. 

Students  who  desire  to  investigate  the  history  of  Algebra  in  more  detail 
should  read  the  second  part  of  Fine^s  Number  System  of  Algebra,  BaWs  Short 
History  oj  Mathematics,  and  Cajori's  History  of  Elementary  Mathematics, 


CHAPTER    XXXII 

APPENDIX. 

L  PROCESSES    ABBREVIATED    BY   USE   OP 
DETACHED    COEFFICIENTS. 

398.  Multiplication  by  Detached  Coefficients.  The  proc- 
ess of  multiplying  one  polynomial  by  another  (see  pp.  42-4) 
can  often  be  much  abbreviated,  and  the  Hkelihood  of  error 
diminished,  by  detaching  the  coefficients  of  the  terms  of  the 
polynomial,  performing  the  multiplication  with  respect  to 
them,  and  then  supplying  the  proper  powers  of  the  letters 
in  the  product  obtained.     Thus, 

Ex.  1.   Multiply  6x'  —  5a;'  -  4x  —  3  by  6a;'  +  5x  —  4. 
Detaching  coefficients, 


6 

-5- 

4- 

3 

6  +  5- 

4 

36 

-30 

-24 

-18 

30 

-25 

-20- 

15 

-  24  +  20  + 

16  +  12 

36  +    0  -  73  -  18  +    1+12 
Annexing  the  powers  of  x,   SQx^  +  Oa:*  -  7Sx^  -  Y^x^  +  a;  +  12, 

or  36a;5  -  nZx^  -  ISa:^  +  a;  +  12,  Product. 

Ex.  2.   Multiply  r'  +  Sa'x  -  2a'  by  x^  -  iax"  +  ^a\ 

1+0+3-2 
1-4+0+  3 


1+0+3-  2 
-4-0-12  +  8 

+  3  +  0  +  9-6 

1-4  +  3-11  +  8  +  9-6 
Hence,  afi  -  4aa;^  +-  ZaH*^  -  lla^^  +  Sa*^;^  +  9a^T  -  Qa^,  Product, 

403 


404  ALGEBRA. 

EXERCISE  138. 

The  pupil  may  work  Exs.  12  to  23  of  Exercise  9  by  use 
of  detached  coefficients. 

As  the  method  is  especially  advantageous  in  multiplying 
polynomials  with  fractional  coefficients,  Exs.  1  to  6  of  Exer- 
cise 13  should  also  be  worked  by  this  method. 

399.  Division  by  use  of  Detached  Coeflacients  is  per- 
formed similarly. 

Ex.  Divide  x'  —  2xY  +  ^xf  —  3^/*  by  x'  +  2xy  —  j^. 

1  +  0-2  +  8-3  11  +  2-1 
1-2  +  3 


1  +  2-1 

-2-1+8 

-2-4+2 

3+6- 
3  +  6- 

-3 
-3 

Hence,  x'^  -  2xy  +  3?/^  Quotient. 

400.  Synthetic  Division.    The  above  process  may  be  fur- 
ther abbreviated  as  follows: 

1+0-2    +8-3|l|-2  +  l,  Divisor  (with  signs 
2  +  1  of  all  the  terms 

+  4    —2  except  the  first 

—  6  +  3  changed). 


Qibotient,  1  —  2  +  3 


+  0  +  0,  Remainder. 


Hence,  x^  —  2xy  +  Zy^,  Quotient. 

This  abbreviation  is  made  possible  by  noticing  that  if  we  change  the 
sign  of  each  term  of  divisor  (in  the  division  in  Art.  399),  we  can  change 
the  successive  subtractions  employed  into  successive  additions. 

Further,  as  each  successive  term  of  the  quotient  is  found  by  dividing 
only  the  first  term  of  the  remainder  by  the  first  term  of  the  divisor,  it  is 
sufficient  to  add  each  column  only  as  needed  in  order  to  determine  the  first 
term  of  each  remainder,  and  hence  the  next  term  of  the  quotient.  It  is 
not  necessary  to  multiply  the  first  term  of  the  divisor  by  each  terra  of  the 
quotient,  since  these  products  are  not  used  in  determining  the  remainders 
which  give  the  successive  terms  of  the  quotient. 

Thus^  in  the  above  process,  having  determined  the  first  term  of  the  quo- 


APPENDIX. 


405 


tient,  1,  we  multiply  —  2  +  1  by  1,  and  add  the  column  _  «  I  the  sum, 

—  2,  divided  by  1  gives  —  2,  the  second  term  of  the  quotient.     We  now 
multiply   -  2  +  1  by  —  2,  set  down  the  product,  +  4  —  2,  in  the  proper 

-2 
place,  add  the  column  +  1,  and  divide  the  sum,  3,  by  1,  etc. 

+  4 
It  is  usually  more  convenient  to  set  the  divisor  in  a  perpendicular  column 
at  the  left.    Thus, 

Ex.  Divide  42^  -  Qx'^y  +  Ax^y^  -  llx'^y^  +  y^  by  ^x^  -  Zxy  -  y\ 


2 
+  3 
+  1 


4-6  +  4-11 
6  +  2 

0+    0 
+    9 


+  0  +  1 


+  3 
-3-1 


+  0  +  0 


2+0+3- 
Hence,  2a;'  +  Sxy^  —  y^,  Quotient. 

EXERCISE   139. 

Solve  Examples  16  to  33  of  Exercise  12  by  the  use  of 
detached  coefficients  and  synthetic  division. 

401.  H.  C.  P.  and  Evolution  by  Detached  Coefficients. 
In  finding  the  H.  C.  F.  of  polynomials  by  the  method  of 
Arts.  119-124,  and  Exercise  38,  the  work  may  be  abbre- 
viated by  the  use  of  detached  coefficients. 

In  extracting  the  square  and  cube  root  of  polynomials  (see 
Arts.  189,  194,  Exercises  68  and  70),  work  may  be  saved  in 
the  same  way. 

It  is  to  be  carefully  noted  that  the  use  of  detached  coef- 
ficients not  only  saves  labor  in  all  these  cases,  but  has  the 
further  advantage  of  diminishing  the  probability  of  mistakes, 
since  fewer  symbols  are  operated  with. 

II.  BEMAINDER  AND  FACTOR  THEOREMS. 
SYMMETRY. 

402.  Remainder  Theorem.  If  any  polynomial  of  the  form 
Pix"" -\r PiQc'' ~ ^  +  PiOf  ~ ^  +  .  .  .  .  -\-pnhe  divided  by x 


remainder  will  be  pid^  +  p-^d" 


-\-  P,a^-'  + 


a^  the 


tained  by  substituting  a  for  ac  in  the  original  expression^. 


406  ALGEBRA. 

Let  the  given  expression  be  divided  hj  x  —  a  till  a  remain- 
der is  obtained  which  does  not  contain  x,  and  denote  the 
quotient  by  Q  and  the  remainder  by  R.     Then 

Pix"" -f-p2x''~'^ -i-p-.iX'''''^  +  ....  +pn^Q(x  —  d) -^  R. 
This  is  an  identity  and  therefore  true  for  all  values  of  x. 

Let  a:  =  rt,  then 

2>,rr+i?2a""'+P3a'*~'+ +Pn  =  Q(ia-ci)  +  R^R. 

.-.  R=p,a''+p,a''-''+psa''-'' +  ....  +  p„. 

Ex.  Find  the  remainder  when  2x^  +  3a;*  —  ox^  +  6x^  +  8a;  —  9 
is  divided  by  x~-2. 

Substituting  2  for  x  in  the  given  expression, 

2  •  2^  +  3  •  2*  -  5  •  2^  +  6  •  22  +  8  •  2  -  9,     or  103,  Remainder. 

403.  Factor  Theorem.  If  any  rational  integral  expression 
containing  cc  become  equal  to  zero,  when  a  is  substituted  for  ac, 
then  ac~a  is  a  factor  of  the  given  expression. 

This  follows  directly  from  the  remainder  theorem  when 
R  =  0,  or  it  can  be  proved  as  follows : 

Let  E  stand  for  the  given  expression.  If  E  be  divided  hy  x  —  a  till  a 
remainder  is  obtained  in  which  x  does  not  occur,  denote  the  quotient  by  Q 
and  the  remainder  by  R.     Then 

E~Q{x-a)  +  R. 
Let  X  =  a,  then 

0  =  Q{0)  +  R  (since  E=0  when  x  =  a). 
.  • .  i2  =  0, 
Hence,  E  =  Q{x  —  a),     or  a:  —  a  is  a  factor  of  E. 

This  principle  is  frequently  of  value  in  factoring  expressions* 

Ex.  Factor  3a;' +  7x^-4. 
By  trial  we  find  that  3x^  +  7a;2  -  4  =  0,  when  x  =  -  1. 

.  •  .  a;  +  1  is  a  factor  of  3a:'  +  la;^  -  4. 
By  division.  3:i^  i-  7x^  -  ^  =  {x  +  1)  {Sx^  +  4a;  -  4) 

=  (a;  +  1)  {x  +  2)  (3a;  -  2),  Factors. 


APPENDIX.  .  407 

It  is  to  be  noted  that  the  only  numbers  that  need  be  tried  as  values  of  x 
are  the  factors  of  the  last  term  of  the  given  expression.  This  follows  from 
the  fact  that  the  last  term  of  the  dividend  must  be  divisible  by  the  last 
term  of  the  divisor. 

EXERCISE  140. 

Factor  by  use  of  the  factor  theorem. 

1.  a;2  — 4.  10.  2:z;2  +  7x  — 15. 

2.  rc2— 3aj  — 28,  11.  2x^  —  x^—lx-^Q. 

3.  x^  —  ^x^2.  12.  4x^  —  4x^—Ux  —  e. 

4.  a^2  —  a\  13.  Sx^  +  Sx^-^'Sx  —  2. 

d.  x^  —  Sa\  14.  2x^  +  x^  —  Ux^  +  5a;  +  6. 

6.  a^—h^  +  3(a  —  h).  15.  6;r*  — 13:z;3  — 45a:2  — 2^:  + 24. 

7.  (a  — 5)2  +  3(a  — 6).       16.x^  —  2x^-{-l. 
S.a^  —  ab\  17.  x^  —  6:^2  +  25. 

9.  a3  +  5a  — 6.  18.  ^4  — 28^2  _^  3S;r  —  90. 

19.  Prove  that  x*^  —  y^  is  always  divisible  by  a?  —  y. 

20.  Prove  that  x^-^-y"^  is  divisible  hyx-\-y  when  n  is  odd. 

21.  Show  that  il  —  x)^is  a  factor  of  l  —  x  —  x''-\-x''+^. 

22.  Show  that  (x— 1)2  is  a  factor  of  na;«+i— (n  + l)ic"  +  l. 

404.  Symmetrical  Expressions.  An  expression  is  sym- 
metrical with  respect  to  two  letters  when  it  is  unaltered  by 
an  interchange  of  the  letters. 

Exs.  a  +  b,     ah,     a'  +  h\     a^  +  ah  +  h\ 

are  each  a  symmetrical  expression  with  reference  to  a  and  h. 

Similarly  an  expression  is  symmetrical  with  respect  to  three 
or  more  letters,  when  it  is  unaltered  by  an  interchange  of  any 
pair  of  them. 

Ex.  a'  +  ^>'  +  c'  — 3a6c, 

is  symmetrical  with  respect  to  a,  6,  c,  since  it  is  unaltered  by 
substituting  a  for  h  and  6  for  a ;  a  for  c  and  c  for  a ;  h  for  c 
and  c  for  6. 
Instead  of  complete  symmetry,  there  are  partial  symmetries  of  different 


408  ALGEBRA. 

kinds  which  an  algebraic  expression  may  have.  Thus,  an  expression  has 
eyclo-symmeb'y  (see  Art.  144,  Ex.  1)  with  reference  to  a,  b,  and  c,  if  it 
remains  unchanged  after  a  is  changed  to  b,  b  to  c,  and  c  to  a. 

Ex.  ab^  +  bc^  +  ea^  has  cyclo-symmetry. 

A  symmetrical  expression  may  often  be  denoted  in  an  abbreviated  way 
by  writing  only  the  typical  terms  of  the  expression  with  the  Greek  letter  2 
before  each  one.     Thus, 

for  a^b  +  b^a    write  I^a^b ; 
for  a^  +  b"^  +  c"^  +  ab  +  be  +  ca    write  2a^  +  2a6. 

Similarly  for  a  product,  as  (a  —  b)  {b  —  c)  (c  —  a), 
write  n{a  —  b). 

405.  Factoring'  S3mametrical  Expressions.  The  factor 
theorem  (Art.  403)  is  frequently  of  use  in  factoring  sym- 
metrical expressions. 

Ex.  1.   Factor  bc(b  —  c)  +  ca(c  —  a)  +  ah{a  —  b). 

When  b  =  c,  the  given  expression  reduces  to 

ca{c  —  a)  +  ac(a  —  c),     or  0. 
Hence,  .  6  —  c  is  a  factor  of  it. 

Similarly         c  —  a    and  a  —  b  are  factors. 
.  * .  bcCb—c)+ca(c—a)+ab(a—b)  =  L(b—c)(c—a)(a~b).  (1) 

Where  L  (since  the  right-hand  member  is  of  the  same 
degree  with  the  given  expression)  is  some  number  to  be 
determined, 

Since  (1)  is  true  for  all  values  of  a,  b,  c, 
Let  a  =  0,     6  =  1,    c  =  2. 

.-.  2(l-2)=:Z(l-2)(2)(-l). 
.-.  X  =  -l, 
and  the  factors  of  the  given  expression  are 
—  (6  —  c)  (c  —  a)  (a  —  6). 


APPENDIX.  409 

Ex.  2.  Factor  a(&+c— a)2  + 6(c+a  — 6)2 +  (•(«  + &—c)^ 
.   H-(6  +  c— a)  (c  +  a— W  (a  +  6— c). 

If  we  put  a=0,  the  given  expression  reduces  to  zero. 

.  *  .  rt,  &,  c,  are  factors  of  it. 

.  *  .  a(6  +  c  — a)2  4-6(c+a— 6)2  +  c(a  +  6— c)2 

+(6+c  —  a)  {c-^a — l)  (a+& — c)=Labc. 

Let  a=l,  &=1,  c=l,  then  i=4. 

.  •  .  Aabc  are  the  factors  required. 

Ex.  3.  Factor  (6'  +  c')  (6  -  c)  +  (c'  +  a')  (c  -  a)  +  (a'  +  6') 
(a -6). 

Evidently  6  —  c,  c  —  a,  a  —  6,  are  factors ;  but  their  product 
is  of  the  third  degree  only,  while  the  given  expression  is  of 
the  fourth  degree.  Hence  the  given  expression  must  contain 
another  factor  of  the  first  degree,  and  since  this  factor  is  sym- 
metrical as  well  of  the  first  degree,  it  must  be  of  the  form 

£(a  +  6  +  c). 
,'.  (b'  +  (f)  (b-c)  +  ((f  +  a')  (c  -  a)  +  (a'  +  6')  (a  -6)  = 

L{b  -  c)  (c  -  a)  (a  +  6)  (a  +  6  +  c). 
Hence,  L  =  l. 

In  factoring  symmetrical  expressions,  it  is  also  useful  to 
remember  (see  Ex.  35  of  Exercise  12)  that 

aM-  6'  +  c'  -  Sabc  =  (a  +  b-\-c)  (a^  +  b'  +  c''-bc  -ca-ab). 
Hence,  for  example, 
ar*  -  82/'  +  27  -f  ISxy  =  r*  +  (-  2yy  +  3'  -  Bx(-  2yX^) 

=  (x-2y  +  SXx'  +  4y'  +  d  +  Qy-Sx  +  2xy). 

EXERCISE  141. 

Show  that^ 

1.  (a  +  6  +  c)'  -  (6  +  c  -  a)'  -  (c  +  a  -  6)'  -  (a  +  6  -  cf  = 
24a6c. 

2.  a'(6-c)  +  6'(c-a)+cXa-6)  =  -(6-c)(c-a)(a-6) 
(a  +  6-i-c). 


410  ALGEBRA. 

3.  a\b  -c)  +  h\c  -  a)  +  c\a  -h)  =  -(h-  c)(c  -  a)ia  -  b), 

4.  (a;  +  2/  +  zy  -  (2/  +  zy  -(z  +  xy  -{x  +  yy  +  X*  +  2/*  +  2*  = 
]  2xyz{x  +  2/  4-  z). 

6.  h'c\h  -  c)  +  cWic  -a)  +  a'b\a  -b)  =  -(b-c)(c-a) 
(a  —  b)  (be  +  ca  +  ab). 

6.  a^h"  -  c')  +  bX(^  -  a')  +  c\a'  —  6^  =  -  (6  +  c)  (c  +  a) 
(a  +  6)(6-c)(c-a)(a-6). 

Factor— 

7.  a;X2/  -  «)  +  2/'(«  -  a;)  +  2"(a;  - 1/). 

8.  a(b  -  c)'  4-  6(c  -  a)»  +  cCa  -  by. 

9.  a(6  -  c)'  +  6(c  -  a)'  +  c<ia  -  6)'  +  8a6c. 

10.  a' +  6' -  c' +  3a6c. 

11.  27  -  Sx'  +  2/'  +  18a:2/'. 


III.  PROOF  OF  THE  BINOMIAL  THEOREM  FOR 
FRACTIONAL  AND  NEGATIVE  EXPONENTS. 

406.  Proof.  In  Arts.  344  and  345  it  has  been  shown  that 
when  n  is  a  positive  integer, 

^^   .     N«     ^    .         .    n(n  —  l)  5  ,  n(n  —  l)(n  —  2)  .  , 
^  1X2  1X2X3 

It  is  evident  that  when  n  is  a  positive  integer  the  number 
of  terms  in  the  right-hand  member  is  limited,  since  all  the 
terms  after  the  (n  +  l)st  contain  n  —  n,  or  0,  as  a  factor.  But 
if  n  is  not  a  positive  whole  number,  the  number  of  terms 
must  be  unlimited,  since  no  factor  becomes  zero. 

Hence  the  binomial  theorem  for  fractional  and  negative 
exponents  gives  an  infinite  series,  and  is  valid  only  when  this 
series  is  convergent.  Hence,  in  the  proof  of  this  theorem 
for  negative  and  fractional  values  of  n,  the  series  used  are 
limited  to  those  cases  where  x  has  such  a  value  as  tvill  make 
the  series  convergent  and  the  proof  is  good  only  for  such  cases. 


APPENDIX.  411 


Denote  Dxe  series 


1x2  1x2x3  •••v; 

by  tlie  symbol  f{n). 

Then,  in  like  manner,  the  symbol  f{m)  stands  for 

1x2  1x2x3  ^  ' 

If  (1)  be  multiplied  by  (2),  the  product  will  be  a  series  in  ascending 
powers  of  x,  whose  coefficients  will  involve  m  and  n  in  a  way  which  is 
independent  of  any  particular  values  wliich  m  and  n  may  have  (just  as, 
for  instance,  (1  +  ax)  (1  +  bx)  EE  1  +  (a  +  b)x  +  abx?'  is  identically  true, 
no  matter  whether  a  and  6  be  whole  numbers  or  fractions,  positive  or 
negative). 

Hence,  to  determine  the  form  which  the  product  must  have  in  all  cases, 
whatever  the  values  of  m  and  n,  we  let  m  and  n  be  positive  integers. 
.  •  ,  f{n)  =  (1  +  xY, 
fim)  =  {1  +  x^. 
Multiplying,  f{n)  x  f{m)  =  (1  +  x^  x  (1  +  a:)«  =  (1  +  »;)•»  +  » 

=  i4-(m  +  r?)a:-f  (^  +  ^H^  +  n-l)^3+.,,. 
1x2 
This  is  the  form  of  the  above  product,  and  holds  for  all  values  of  m 
and  n. 

.'.  f{m)  xfin)=f{m  +  n) (3) 

Similarly, 

/(m)  x  /(n)  xf{p)^  f{m  +  n)  x  f{p) 
=  f{m  -\-  n  +  p), 
And  /(m)  x  f{n)  x  f{p)  ...  .to  s  factors  =f{m-\-n  +p  +  ....  to  s  terms). 

Let  m  =  71  =  p  =  ....  =  — >  where  r  and  s  are  positive  integers. 

s 

Then  /(-)  •  f(-\ tosfactors=/(-  +  -+-  + ....to s terms). 

•  ■•[/(';)]"-/« (4) 

But  since  r  is  a  positive  integer,  /('•)  =  (!  +  x)^. 

Substitute  for/(r)  in  (4),  (1  +  xY  =  [f(^)T' 

Extract  .sth  root,  {I  +  x)^  =f(^\'  ' 


412  ALGEBRA. 

This  proves  the  binomial  theorem  for  any  positive  fractional  exponent. 
We  can  now  prove  that  the  theorem  is  also  true  for  any  negative  expo- 
nent. 
In  (3)  let  m  =  -  n. 

.•./(-n)x/(n)=/(-n  +  n)=/(0)  =  l. 

•'^       ^      fin)       {1  +  xr      ^  ^ 

.'.{l  +  x)--=f{-n) 

^        ^  1x2 


IV.   A  MISCELLANEOUS  EXERCISE. 
EXERCISE    142. 

Factor — 

1.  lOOOa;'— y'.  5.  132x -{- xy  —  xy\ 

2.  36a:'  — 13a;2  — 40a;.  6.  4a;*  +  4a''  — a*— 4. 

3.  36a;*— 289a;2+400.  7.  a;^  — 9. 

4.  a;^  — 64a;.  8.  a;^  +  27. 

13.  15aa;  — 5ai/  +  12&a;  — 4&I/. 

14.  7(p-l)»-27(p  — 1)  +  18. 

15.  a^  +  Q  — 4a;'^  — w'^  +  4na;  — 6a. 

16.  2aV4-(a'+3a)a;  — a-  — 3a  — 2. 


9. 

4x  —  y\ 

10. 
11. 

a;2« -!/-«. 
J-8b-K 

12. 

25n^—y-\ 

17. 

3a;_8a;^— 35. 

21. 

a*a;^-3a^  +  5a;^— 15. 

18. 
19. 
20. 

6a;3-a;"''-15. 
lOar^— 19a;*  — 56. 
12a;^  +  5a;«  — 72. 

22. 
23. 

24. 

30-\-V'2x  —  2x. 
60  — 7-1/ 3a  — 6a. 
15x  —  2-\/xy  —  2^y, 

ltx  =  2,  y  =  — 3,  z  =  —  i,  find  the  value  of: 

25.  3a;i/  — 2/(a;  +  4^)  — a;0(4i/  +  6a;)  +  3?/^(a;  +  i/)  {y-\-2z). 

26.  {x  —  y  —  lO^r)  (2a;  -\-3y  +  62)  —  xyz  (x'  —  y'  —  20^)  +  UxY^^, 

27.  {y^  —  z){x  —  5)-{-{x'-\-2)  {y  —  20)-^{2x-{-5y—lO0)\ 

28.  x'^y  +  xy^  +  xz""  +  x'z  +  y^z  +  yz^  —  3yz{y^  —x  —  1). 

29.  a^^yx-{.z^y  ^  (^x  -^  y^-y  —]/^—{xz  +  xy)^  —  {y  -\-  z)x' 


APPENDIX. 


412 


Find  the  H.  C.  F.  and  the  L.  C,  M.  of:  ' 

30.  a?-\-ah^  ^-a^h^—y  and  a^  —  ^V-aV  — b', 

31.  a'  +  6  -/ a  —  ah  +  i/¥  and  a" -\- a  ^~ab  -\-ab^  —  h  i/X 

32 .  a^+  &  i/"a  —  26^  and  a^  +  «&*—&  i/"^  —  &^. 

33.  3a2  +  7aM  +  a^h^  —  V  and  2a2—  13a6  —  2a^&^  +  3&'. 

Simplify: 

34.  (i/"^— 1)  (3i/^+2)+a;^(2i/'^  — 5)  — (2i/^  — 3)(3a;^  — 4). 


85. 


+  w  ,  m  —  n 


m  —  n      m-\-n ^__^ 

m  -\-  n      m  —  n       n     m 


m  —  u      m 


36.  \ 


x—\ 


x  + 


x  —  \- 


x+l 


x  +  1 


37.   8"^  +  25^— (D-'  +  lS^  — (^)    I 


h^l/ab 


27a2b- 


■^{x  +  D. 


39.  -/|-+i/|_-/i  +  ^36; 


40.   l/^-2l/j  +  3l/6^ 
1/2  —  21/3  —  3.1/6* 

Solve: 
41.  dx''  —  2x{x-{-5)—{x  —  l)  (2a; +  3)  =  (5a;  — 1)  (a;  +  7)  — 6a;». 

3a;— 2      7      a^  —  1  ^       2 
'   2a;— 3      6      a;  +  2  3* 

43.  3j;  +  lli/  =  19|  11a;  — 32/  =  9, 


414  ALGEBRA. 

U.  2x  —  6y  —  2  =  5.  47.  «a;-f  (&  +  l)y  =  c. 
3x  +  y—y=6,  {b-{-i)x  +  ay=d, 

^x  —  9y-{-2  =  23. 

..  ^    K     10  ^®-  ii  +  iy  =  -'^' 

45.  ^-52^  =  13.  _4.  ,  -£      „ 

^  +  i.=  4.  ^^  +  «^  =  '- 

46.4^-§  =  -3.  49    a.-f5,-fc.  =  l. 

f  bx-i-  cy  -\-az  =  l, 

dx  -\-  4.y  =  —  6i.  ex -\-  ay  ~{- b0  =  1. 


Write  the  following  expressions  with  positive  exponents 
and  reduce  the  results  to  single  fractions. 

50.  a  +&-'  +  2C&-2.  ^^    4(x  —  l)-^-^x-Hx-l). 

51.  (a  — &)-i +  («+&)-'.  '  {x~l)-'  —  x-^ 

52.  a(a  —  &)-i  — &(«  +  &)-'. 


_^^  ,   1_-^  55.  a-'  +  Va-'b-^-i-h- 


56. 


(a  +  &)  (a  — &)-i  — (a  — &)  {a-\-b)-^ 


l-(a2H-6^)(a  +  6) 


— 2 


59      (a-^  +  &-^-c-M' 
■  a-'''&-na  +  &)'  — c-2' 

60.  a(l -«-•'»)  (a  +  a-i)(l  +  a-i)-Ma'  +  l)-^ 

62.  6(a  +  &)-i  — a&(a  +  &)-2  — a&2(^4.j)_3^ 

63.  (&  +  <?)(«  — &)-U«  —  c)-i+(c  + a) (&  —  c)-i(&—a)-i  + 

(a4-6)  (c  — a)-Hc  — &)-!. 


64    (l  +  ^)(l  +  ^')-^-(l+^^)(l  +  ^)~^ 


APPENDIX.  415 

65.  ar^^-    ar^^"!'    x^-^.  69.  z""-^'    ;^»+«-5-(;?2)*. 

66.  (a;«-^)«+^-Ha;-^^  70.  (rr-)""'-  W"^+'-  (a;-)'-^ 

67.  x^'  +  «^  -f-  a;2«6.  71.  (a;«)«+^.  (a;^)'*-^--  (a;«+2&)^ 

68.  (a;--!)"""'-    (.;-)"*.  72.  {y-^Y'  {y^-'Y -i-{t^+^)-\ 

73.  ^3m.  — 71.     ^2n  +  l.    ^m  +  2    .    j,2m  +  n  +  3^ 


g  1 

75.   ^a  +  l.     ^a+1 


a;a+l_|_|/a;-\  78. 


l-a*&*         (l  +  a*&*)' 


6*v''«  — «&  «      ^ 


77 


Solve  the  following  simultaneous  quadratics: 

81.  a;*  4- 2/*  =  272.  88.  a;-i  +  2/-'=4. 

a;  +  i/  =  6.  a;-2  +  ^-2  =  8i. 


82.  T>  —  y^  =  ^3. 

x-y  =  3.  «^-a-^r6  =  *- 


89.^^^- 


xy  +  y^  =  18.  3x^4y~^^- 


x  —  y  =  5 


y  2  u  1 

y^  —  xy  —  da;  =  —  1, 


85.  3x'—lxy-\-y''—4:X=^ld^  91.  2x'~lxy -\-^y=  —  ^^ 

7xy  —  y^  =  —15.  6x  —  4y=15. 


86.  xhj'\'xy^=6. 


X  2y 


3xY  +  8xy  =  3.  ^2.^  +  ^  =  1. 

87.x^  +  y^  =  5.  ^4-^=2 

ic4-y  =  35.  2'^3 


416  ALGEBRA. 


.,+i+L  =  3i. 


^«-Af+V!=- 


94.  a;  +  2/  +  v^ary  =  14. 
VVy{x-Jry)  =  ^0.  x-\-y  =  \Q. 

95.  i/^M^7  +  y  =  6.  99.  V'^-V^^y  =  ll, 
1/FT2V  +  ^'  =  22.  -i^^ - 2/  Vxy  =  60. 

96.  i/Ff^  +  V^^^  =  4.  100-  y  +  l/^'  —  1  =  2. 


I.  Transformations  of  Physical  Formulas. 

101.  Given  v=at,  find  the  value  of  t  in  terms  of  a  and  v 

102.  Given  s  ==  ^at^,  find  the  value  of  t  in  terms  of  a  and  s. 

103.  Given  s  =  — ,  find  the  value  of  v  in  terms  of  a  and  5. 

104.  Given  s  =  |a  (2^  —  1),  find  ^  in  terms  of  a  and  s. 


mv 


105.  Given  /^  = ,  find  each  letter  in  terms  of  the  other  three. 


r 


106.  Given  e  =  -^,  find  each  letter  in  terms  of  the  others. 

uw  ^ 

107.  Given   e=-?r— ,  find  each  letter  in  terms  of  the  others. 

2a 

108.  Given    t=7:y  —,  find  I  and^,  each  in  terms  of  the  other  letters. 

109.  Given  0=77,  find  each  letter  in  terms  of  the  others. 

li 

OS 

110.  Given  R  =  — — ,  find  each  letter  in  terms  of  the  others. 

9  +  s' 

111.  Given  -7  =  —  -\ — •.,  find  each  letter  in  terms  of  the  others. 


APPENDIX,  417 

n.  Transformations  of  Arithmetical  Formulas. 

112.  Given  i=prt,  find  each  letter  in  terms  of  the  other  three. 

113.  Given  a=p  +  prt,  find  each  letter  in  terms  of  the  other  three. 

m.  Transformations  of  Algebraic  Formulas. 

114.  Consult  pages  318  and  326. 

IV.  Transformations  of  Geometrical  Formulas. 

115.  Given  A  =^bh,  find  each  letter  in  terms  of  the  others. 

116.  Given  A  =^h(b  +  b'),  find  each  letter  in  terms  of  the  others. 

117.  Given  C=27tR,  find  each  letter  in  terms  of  the  others. 

118.  GivenA=7^JR^    find  each  letter  in  terms  of  the  others. 

119.  Given  A  =7tRL,  find  each  letter  in  terms  of  the  others. 

120.  Given  A  =47zR^,  find  each  letter  in  terms  of  the  others. 

121.  Given  T  =7zR{R  +  L),  find  each  letter  in  terms  of  the  others. 

122.  Given  T  =27zR(R  +  II) ,  find  each  letter  in  terms  of  the  others. 

123.  Given  V=7rR^H,  find  each  letter  in  terms  of  the  others. 

124.  Given  V^^ttR^II,  find  each  letter  in  terms  of  the  others. 

125.  Given  V=^7:R^,  find  each  letter  in  terms  of  the  others. 

126.  Given  a+b+c=2s. 

show  that  a  +  b—c=2(s—c);    a—b  +  c=2(s—b),  etc. 

.««     .,        r,        xu  X  -.      a'  +  b^-c^     2(s-a)(s-b)  ,  .t,  * 

127.  Also  show  that  1 -^ =  "^ ab  '  ^* 

,     a^-\-c^-b^  _2(s-a)(s-c)  ^ 

1 —       ^ ,     etc. 

2ac  ac 

128.  Also  show  that  1+^^=^=?^^^;    and  that 

a2fc2-62     2s(s-b) 

H n =-^ >    etc. 

2ac  ac 


CHAPTER    XXXIII. 
GRAPHS. 

407.  Definitions.  A  variable  is  a  quantity  which  has 
an  indefinite  number  of  different  values. 

A  function  is  a  variable  which  depends  on  another 
variable  for  its  value. 

Thus,  the  area  of  a  circle  is  a  function  of  the  radius  of  the  circle; 
the  wages  which  a  laborer  receives  is  a  function  of  the  time  that  the  man 
works. 

A  graph  is  a  diagram  representing  the  relation  between 
a  function  and  the  variable  on  which  the  function  depends 
for  its  value. 

A  function  may  depend  for  its  value  on  more  than  one  variable; 
thus  the  area  of  a  rectangle  depends  on  two  quantities — the  length  of 
the  rectangle  and  the  breadth.  The  present  treatment  of  graphs,  how- 
ever, is  hmited  to  functions  which  depend  on  a  single  variable. 

408.  Uses  of  Graphs.  A  graph  is  useful  in  showing 
at  a  glance  the  place  where  the  function  represented  has  the 
greatest  or  least  value,  where  it  is  changing  its  value  most 
rapidly,  and  in  making  clear  similar  properties  of  the  func- 
tion. 

Graphs  of  algebraic  equations  are  useful  in  making  plain 
certain  properties  of  such  equations  which  are  otherwise 
difficult  to  understand.  A  graph  also  often  furnishes  a  rapid 
method  of  determining  the  root  (or  roots)  of  an  equation. 

Copyright,  1906,  by  Fletcher  Durell. 

418 


GRAPHS. 


419 


409.  Framework  of  Reference.  Axes  are  two 
straight  lines  perpendicular  to  each  other  which  are  used 
as  an  auxiliary  framework  in  constructing  graphs,  as  XX' 
and  YY'.  The  x-axis,  or  axis  of  abscissas,  is  the  hori- 
zontal axis,  as  XX'.  The  y-axis,  or  axis  of  ordinates, 
is  the  vertical  axis,  as  FF'. 


The  origin  is  the  point  in  which  the  axes  intersect,  as  the 
point  0.  The  ordinate  of  a  point  is  the  line  drawn  from 
the  point  parallel  to  the  ^/-axis  and  terminated  by  the  x-axis. 
The  abscissa  of  a  point  is  the  part  of  the  x-axis  inter- 
cepted between  the  origin  and  the  foot  of  the  ordinate. 
Thus,  the  ordinate  of  the  point  P  is  AP,  and  the  abscissa 
is  OA. 


abscissa,  the  "x"  of  a  point. 

Ordinates  above  the  x-axis  are  taken  as  plus,  those  below, 


420 


ALGEBRA. 


as  minus;  abscissas  to  the  right  of  the  origin  are  plus,  those 
to  the  left  are  minus. 

The  co-ordinates  of  a  point  are  the  abscissa  and  the 
ordinate  taken  together.  They  are  usually  written  to- 
gether in  parenthesis  with  the  abscissa  first  and  a  comma  be- 
tween. 

Thus,  the  point  (2,  4)  is  the  point  whose  abscissa  is  2  and  ordinate 
4,  or  the  point  P  of  the  figure.  Similarly,  the  point  (  —  3,  2)  is  Q; 
(-2,  -2)  is  R;  and  (1,-4)  is  S. 

The  quadrants  are  the  four  parts  into  which  the  axes 
divide  a  plane.  Thus,  the  points  P,  Q,  R,  and  S  lie  in  the 
jirstj  second,  thirds  and  fourth  quadrants  respectively. 

+Y 


Q 


X\        I        I 


O 


Rl 


P 

--1(2,4) 


■i — f— iZ, 


s 

(1,  -4) 


F 


EXERCISE  143. 

Draw  axes  and  locate  each  of  the  following  points: 
1.  (3,2),  (-1,3),  (-2,-4),  (4,-1). 


GRAPHS.  421 

2.  (2,§),  (-3,-li),  (5,-f),  (-2,J). 

3.  (2,  0),  (-3,  0),J0,  4),  (0,-iMO,  0).  _ 

4.  (l,\/2),  (1,-V2)  (Vs;  0)  (V5,-3),  (-iV5,  2\/2). 

5.  Construct  the  triangle  whose  vertices  are  (1, 1),  (2,-2), 

(3,  2). 

6.  Construct  the  quadrilateral  whose  vertices  are  (2,-1), 

(-4,-3),  (-3,5),  (3,4). 

7.  Plot  the  points   (0,  3),  (1,  3),  (2,  3),  (5,  3),  (-1,  3), 

(-2,  3),  (-5,3). 

8.  Also  (0,  0),   (1,  0),  (2,  0),  (5,  0),  (-1,  0),  (-3,  0), 

(-5,0). 

9.  Also  (0,  0),  (0,  1),  (0,2),  (0,  3),  (0,  5),  (0,  -1),  (0,  -3), 

(0,  -5). 

10.  All  points  on  the  a:-axis  have  what  ordinate? 

11.  All  points  on  the  i/-axis  have  v/hat  abscissa? 

12.  Construct  the  rectangle  whose  vertices  arc  (1,  3),  (6,  3), 

(1,  -2),  (6,  -2)',  and  find  its  area. 

13.  Construct  the  rectangle  whose  vertices  are  (  —  3,  4), 

(4,  4),  (-  3,  -2),  (4,  -2),  and  find  its  area. 

14.  Construct  the  triangle  whose  vertices  are  (-3,  -4), 

(  -1,  3),  (2,  -4),  and  find  its  area. 

15.  In  which  quadrant  are  the  abscissa  and  ordinate  both 

plus  ?  both  minus  ?  In  which  quadrant  is  the  ab- 
scissa minus  and  the  ordinate  plus  ?  In  which  is 
the  abscissa  plus  and  the  ordinate  minus  ? 


G-RAPHS    OF    EQUATIONS    OF   THE   FIRST 
DEGREE. 

410.  To  construct  the  graph  of  an  equation  of  the 
first  degree  containing  two  unknown  quantities,  as 
X  and  y,  let  x  have  a  series  of  convenient  values,  as  0,  1,  2, 
3,  etc.,  —1,  —2,  —3,  etc.;  find  the  corresponding  values  of  y; 


422 


ALGEBRA. 


locate  the  points  thus  determined    and  draw  a   line   through 
these  points. 

Ex.     Construct  the  graph  of  the  equation  y  =  2x—l. 


I,et  2-=     1      0  1     1     1     2     1     3          etc.       |     -1     |     -2 

etc. 

Theny=      -1        1     |     3         5     |      etc.       |     -3 

—  5          etc. 

Construct  the  points  (0,  -1),  (1,  1),  (2,  3),  (3,5),  (-1,  -3),  (-2,  -5), 
etc.,  and  draw  a  line  through  them.  The  straight  Hne  AB  is  thus 
found  to  be  the  graph  of  y  =2x—l. 


F 

t 

J 

t 

f 

t 

1 

^                                    0 

t 

1 

t 

7 

i_ 

7 

4- 

Y' 

'X 


411.  Linear  Equations.  It  will  always  be  found  that 
the  graph  of  an  equation  of  the  first  degree  containing  not 
more  than  two  unknown  quantities  is  a  straight  line.    Hence 

A  linear  equation  is  an  equation  of  the  first  degree. 

412.  Abbreviated  Method  of  Constructing  the  Graph 
of  a  Linear  Equation.      Since   a  straight    line    is    deter- 


GRAPHS. 


423 


mined  by  two  points,  in  order  to  construct  the  graph  of  an 
equation  of  the  first  degree  it  is  sufficient  to  construct  any 
two  points  of  the  graph  and  to  draw  a  straight  line  through 
them. 

The  greater  the  distance  between  the  points  chosen,  the  more  accurate 
the  construction  will  be.  It  is  usually  advisable  to  test  the  result  ob- 
tained by  locating  a  third  point  and  observing  whether  it  falls  upon 
the  graph  as  constructed. 

If  the  given  line  does  not  pass  through  the  origin,  or  near  the  origin 
on  both  axes,  it  is  often  convenient  to  construct  the  line  by  deter- 
mining the  points  where  the  line  crosses  the  axes. 

Ex.  1.     Graph3?/-2x  =  6. 

When  X  =0,  t/=2;  when  t/=0,  x  =S.  Hence  the  graph  passes 
through  the  points  (0,  2)  and  (-3,  0),  or  CD  is  the  required  graph. 

lY 


t-H — I     I     iX 


Ex.  2.  Graph  4x  +  7i/  =  1. 

When  x  =  0,  ?/  =  I ;  when  y  =  0,  x  =  i.  Hence  the  graph  passes  close 
to  the  origin  on  both  axes.  Hence  find  two  points  on  the  required 
graph  at  some  distance  from  each  other  as  by  letting  x  =  0,  and  9  and 
finding  ij  =  \,  —5.     Let  the  pupil  construct  the  figure, 


424  ALQEBBA, 


EXERCISE  144. 

Graph  the  following  (it  is  an  advantage,  if  possible,  to  draw 
the  graph  line  in  red,  the  rest  of  the  figure  in  black  ink) : 

1.  y  =  x-\-2.  8.  y^-x. 

2.  y  =  x-2.  9.  i/  =  4. 

3.  3a; +  2?/ =  6.  10.  If     x  =  2,     show     that 

4.  y  =  2x.  whatever  value  y  has,  x  al- 

5.  4:X  —  5y=l.  ways  =  2.      Hence  the  graph 
X— 1      „  ofa;  =  2isa  line  parallel  to 

^'     2     ~    ^*  the  X  -  axis. 

7.  x=3(y—l).  11.  Graph  x  =  0;  also  y=0. 

12.  Show  how  to  determine  from  an  inspection  of  a  linear 
equation  whether  its  graph  passes  through  the  origin;  near 
the  origin  on  one  axis;  near  the  origin  on  both  axes. 

13.  Graph  5x  +  Qy  =  l;  also  6x-y  =  12. 

14.  Obtain  and  state  a  short  method  of  graphing  a  linear 
equation  in  which  the  term  which  does  not  contain  x  or  y  is 
missing,  as  2y—Sx=0. 

Before  graphing  the  following  determine  the  best  method 
of  constructing  each  graph,  and  then  graph: 

15.  x  +  2y  =  4:.  19.  ^x  +  iy  =  i.  23.  x~y  =  5. 

16.  2y  =  x.  20.  x=~S.  24.  y-\-2  =  0. 

17.  5x~ey  =  l.  21.  5x  +  4y  =  0.  25.  Sx-2y  +  \^=0. 

18.  yi-3x  =  0.  22.  8x+3y  =  2.  26.  4x  =  12 

27.  Construct  the  triangle  whose  sides  are  the  graphs  of 
the  equations,  y  —  2x  +  l=0,  Sy  —  x~-7  =  0,  y  +  3x+ll  =0. 

28.  Construct  the  quadrilateral  whose  sides  are  the  graphs 
of  the  equations,  x—2y-4:  =  0,  x+y  =  l,  Sy  — 5x-15  =  0, 
x+2y-4:  =  0. 

29.  An  equation  of  the  form  y  =  h,  represents  a  line  in 
what  position  ?     One  of  the  form  x  =  a? 


GRAPHS. 


425 


413.  Graphic  Solution  of  Simultaneous  Linear 
Equations.  If  we  construct  the  graph  of  the  equation 
x  —  y  =  S  (the  hne  AB)  and  the  graph  of  3x-\-2y  =  4  (the 
line  CD),  and  measure  the  co-ordinates  of  their  points  of  in- 
tersection, we  find  this  point  to  be  (2,-1). 


Y 

c> 

\ 

\ 

\ 

\ 

\ 

/B 

\ 

/ 

\ 

/ 

\ 

\ 

/ 

0 

\ 

&■ 

/ 

1) 

/ 

S. 

/ 

\ 

/ 

\ 

/ 

\ 

\ 

a' 

/> 

Y' 

{/r t/=  3 
3x-l-2w=^4 
by  the  ordinary  algebraic  method  we  find  that  x  =  2  and 
2/=  —1.  In  general,  the  root^  of  two  simultaneous  linear 
equations  correspond  to  the  co-ordinates  of  the  point  of  inter- 
section of  their  graphs.  For  these  co-ordinates  are  the  only 
ones  which  satisfy  both  graphs,  and  their  values  are  also  the 
only  values  of  x  and  y  which  satisfy  both  equations. 

Hence,  to  obtain  the  graphic  solution  of  two  simul- 
taneous equations,  draw  the  graphs  of  the  given  equa- 
tions, and  measure  the  co-ordinates  of  the  point  {or  points)  of 
intersection. 


426 


ALGEBRA. 


In  certain  cases  (as  when  the  values  of  x  and  y  are  not  inte- 
gral, or  when  the  graphs  have  already  been  constructed),  the 
graphic  solution  of  a  pair  of  equations  is  more  convenient 
than  the  algebraic  solution.  In  making  graphic  solutions  of 
equations  whose  roots  are  not  integral,  cross  section  paper  in 
which  each  linear  unit  has  been  subdivided  into  five  or  ten 
equal  parts  should  be  used. 


Ex.     Solve 


(1). 


,.     „     (5x-{-Sy  =  2.. 
graphically  I  2^_^J^g^__  ^^2). 


Constructing  the  graphs  of  equations  (1)  and  (2)  and  measuring  the 

■0-92+       Uoote. 


co-ordinates  of  their  point  of  intersection,  -j         J  q  ^ 


\ 


^ 


^ 


zkd 


<2i 


^ 


U 


414.  Special  Case.   Simultaneous  Linear  Equations 
whose  Graphs  are  Parallel  Lines.     Let  the  pupil  con- 

{x-\-2y  =  2 
3^1  2^,^10 


OBAPHS,  427 

He  will  find  that  the  graphs  obtained  are  parallel  straight  lines. 
Let  him  now  try  to  solve  the  same  equations  algebraically.  He  will 
find  that  when  either  x  or  y  m  eliminated,  the  other  unknown  quantity 
is  eliminated  also,  and  that  it  is  therefore  impossible  to  obtain  a  solu- 
tion. The  reason  why  it  is  impossible  to  obtain  a  solution  is  made 
clear  by  the  fact  that  the  graphs,  being  parallel  lines,  cannot  intersect; 
that  is  to  say,  there  are  no  values  of  x  and  y  which  will  satisfy  both  of 
these  lines,  or  both  equations,  at  the  same  time. 

415.  Graphic  Solution  of  an  Equation  of  the  First 
Degree  of  One  Unknown  Quantity.     By  substituting 

for  y  in  the  first  equation  of  the   pair]  ^~q~   ;  the  two 

equations  reduce  to  a:  — 3  =  0.  Accordingly,  the  graphic  solu- 
tion of  an  equation  like  x  —  3  =  0  can  be  obtained  by  combin- 
ing the  graphs  of  y  =  x  —  3  and  y=0.  In  other  words,  the 
root  of  x  — 3  =  0  is  represented  graphically  by  the  abscissa 
of  the  point  where  the  graph  of  y=x—3  crosses  the  x-axis. 

EXERCISE  145. 

Solve  each  pair  of  the  following  equations  both  graphically 
and  algebraically,  and  compare  the  results  in  each  example: 


1. 


(2x  +  Sy  =  7,  .      {x  +  7y+n  =  0. 

\x-y=l,  ^'    \x-3y+l=0. 

2     (y  =  Sx-4.  (y  =  3. 

^-    Xy=-2x+l.  ^'    \9x-5y^S, 

^-    Xx  +  y  +  Q  =  0.  ^'    l2/  =  2x  +  3. 

.     (y  =  2x,  8.  Solve  graphically,  2a;+ 3  =  0. 

\x+y=0.  g   Also3a:-5  =  0. 

10.  Discover  and  state  the  relation  between  the  coefficients 
of  two  linear  simultaneous  equations  whose  graphs  are  par- 
allel lines. 


428  ALGEBRA, 

11.  Solve  graphically  j 


Sx+by=7. 


Qx-\-2y=n. 

{Ox '\y  =  5 
fi    _q    _c* 


GRAPHS   OF   QUADRATIC   AND  HIGHER 
EQUATIONS. 

416.  To  construct  the  graph  of  a  quadratic  equation 
of  two  unknown  quantities,  use  the  method  of  Art.  410. 
Sometimes,  however,  it  will  be  found  advantageous  also 
to  let  X  have  fractional  values  as,  J,  i,  yV;  h  ^tc.     The  ob- 


servant pupil  will  also  find  methods  of  abbreviating  the  work 
in  certain  cases. 

It  will  be  found  that  the  graph  of  a  quadratic  equation  of 


ORAFES. 


429 


two  unknown  quantities  is,  in  general,  a  curved  line,  and,  in 
particular,  either  a  circle,  parabola,  ellipse,  or  hyperbola. 
Ex.  1 .     Construct  the  graph  of  2/  =  a:^  —  3x  +  2. 


Let    x=  1    0       12 

3    1    4    1     f         etc.       -1      -2  1    etc. 

Then  y  = 

2       0       0 

2       6 

-\  1    etc.           6       12  j     etc. 

The  graph  obtained  is  the  curve  ABC.  A  curve  of  this  kind  is  called 
a  Parabola.  The  path  of  a  projectile,  for  instance  that  of  a  baseball 
when  thrown  or  batted  (resistance  of  the  air  being  neglected),  is  an 
inverted  parabola. 

Ex.  2.     Construct  the  graph  of  4a;2  —  Q^/^  =  36. 


Let     a;=0                         1                 2            3        4           5          6     etc. 

Theny=  ±2\/-l    ±  ^V-2    ±  fV-5  |  0      ±1.7    ±2.6    ±3.4!etc. 

For  negative  values  of  x  the  values  of  y  are  the  same  as  for  the  cor- 
responding positive  values  of  x.  Hence  the  graph  is  a  curve  of  two 
branches,  ABC  and  A'B'C,  of  the  species  known  as  the  Hyperbola. 


Y 


a: 


^n: 


^v 


'X- 


B' 


:z: 


430 


ALGEBRA. 


EXERCISE  146 

Graph  the  following: 
1.  y  =  x^—\. 


2.  y  =  x^-2x-'i. 

3.  y=\x'. 

4.  y'^  =  4:X  —  x^. 

6.  1/2  =  4a;. 

7.  a;2-2/2  =  9. 


8.  xy  =  4:. 

9.  a:  +  a:?/=l. 

10.  a:2+(?/-4)2  =  5. 

11.  92/2-a:2=-9. 

12.  y^  =  4:X  +  4:. 

13.  a:2-a:?/  +  T/2  =  25. 

14.  2/2  =  4. 


15.  x2  — 4a;  +  3  =  0  (show  that  whatever  the  value  of  y, 
X  always  =1  or  3;  hence  the  graph  is  two  straight  lines  paral- 
lel with  the  2/-axis) . 

417.  G-raphic  Solutions  of  Simultaneous  Equations 
Involving  Quadratics.  The  method  employed  is  stated 
in  general  in  Art.  413. 

Y 


GRAPHS,  431 


Ex. 


{t2  -L  7/2  _  25 


Constructing  the  graph  of  x^+y^  =25,  we  obtain  the  circle  ABC, 
Constructing  the  graph  of  x+y  =1,  we  obtain  the  straight  hne  FH. 

Measuring  the  co-ordinates  of  the  points  of  intersection  of  the  two 
graphs,  the  points  are  found  to  be  (4,  —3)  and  (  —  3,  4).  These  re- 
sults may  be  verified  by  solving  the  two  given  simultaneous  equations 
algebraically. 

418.  Special  Cases.    Imaginary  Roots.    Let  the  pupil 

(   'T»2  _|_  nj2  ——  A 

construct  the  graphs  of  ^      t     _  q    •      He  will  find  that  these 
(.  x-j-y  —  o 

two   graphs  do   not    intersect.       Let  him  then   solve    the 

given  equations   in   the   ordinary   algebraic  way.      He  will 

find  that  the  roots  are  imaginary.     If  he  treat  the  equa- 

(  a;2    r   2,2  _  1 

tions  •]  .  2  j_  Q  2  _  Q A  ill  the  same  way,  he  will  obtain  a  simi- 
lar result.  In  general,  imaginary  roots  of  simultaneous  equa- 
tions correspond  to  points  of  non-intersection  of  the  graphs  of 
the  given  equations. 

It  should  be  remembered  that  in  solving  a  pair  of  simultaneous 
equations,  the  number  of  values  of  x  (and  also  of  y)  is  equal  to  the 
sum  of  the  degrees  of  the  two  equations.  Hence,  if  two  simultaneous 
equations  are  both  of  the  second  degree,  their  graphs  should  intersect 
in  four  points;  and  if  their  graphs  are  found  to  intersect  in  only 
two  points  for  instance,  the  other  two  points  must  correspond  to 
imaginary  roots.    The  pupil  may  illustrate  this  by  graphing  and  also 

solving  algebraically  j  ^2^^2=5. 

EXERCISE  147. 

Solve  both  graphically  and  algebraically: 
■         (0:2  +  2/2  =  25.  o     j  32/2-2x2  =  12. 

^-    \x-y=l.  ^-    \x^  +  y^  =  lQ. 

o     jx+2/=-2.  4     (3x2  +  2/2=a 

^'    |an/=-3.  *•    iy=x+2. 


432  ALGEBRA,       . 

^*    (32/2  +  2;r2=14.  ^'    \x^  =  dy-y^. 

.     ja:2  +  2/2-10a;=0.  «      (4x2-92/2  =  36. 

^*    (2/-=2x.  ^-     (^^  +  2/^  =  1- 

^     (0:2  +  2/2==  16.  .^     Jx2  +  2/2  +  x  +  32/=18. 

'•    (x2  +  4?/2-43.  \xy-y=l2. 

419.  Graphic  Solution  of  a  Quadratic  or  Higher 
Equation  of  One  Unknown  Quantity.     By  substituting 

for  y  in  the  first  equation,  tbxC  pair  of  equations  i     _  n 

reduces  to  ^2  — 3x+2  =  0.     Accordingly,  the  graphic  solution 
of  an  equation  like   x2  — 3x  +  2  =  0  is  obtained  by  solving 

graphically  the  pair  j  ^~^        '*' 

In  other  words,  the  roots  of  a  quadratic  equation  of  one  un- 
known quantity,  ax^  +  bx  +  c  =  0,  are  represented  graphically  by 
the  abscissas  of  the  points  where  the  graph  of  y  =  ax^-{-bx-\-c 
meets  the  x-axis. 

Ex.     Solve  graphically  ^2  —  3a:  +  2  =  0. 

The  graph  of  y  =x'^  —  ^x-\-2  is  the  curved  line  ABC  of  the  figure  in 
Art.  416  (p.  428). 

This  curve  crosses  the  x-axis  at  the  points  (1,  0)  and  (2,  0). 

.'.  a:  =1,  2,  Roots. 

The  same  results  are  obtained  by  solving  the  equation  x^  — 3a:-|-2  =0 
algebraically. 

This  method  of  solution  also  applies  to  a  cubic  equation  or 
to  an  equation  of  one  unknown  quantity  of  any  degree. 

Thus  to  solve  the  equation  x'  —  Sa:"  +  Sx  —  2  =  0,  graph  the  equa- 
tion y  ==  x^  —  3x^  +  5x  —  2.  The  abscissas  of  the  points  where  this 
graph  crosses  the  x-axi«5  have  the  same  value  as  the  roots  of  the  given 
equation  x^  -  Sx^  -\-  5x  -  2  =  0. 

430.  Special  Cases.  Let  the  pupil  construct  the  graph 
of  each  of  the  following: 


0RAPH8.  433 

y=x'^-2x-l....{\) 
y  =  x^-2x+l....{2) 
2/=x2-2a:+2....(3) 

It  will  be  found  that  the  graph  of  (1)  crosses  the  x-axis  at  two  points, 
and  that  x'^  —  2x—\  =0  therefore  has  two  real  and  unequal  roots;  that 
the  graph  of  (2)  meets  the  a:-axis  at  only  one  point,  and  that  accordingly 
the  equation  x'^  —  2x-\-\  =0  has  two  real  and  equal  roots;  and  that  the 
graph  of  (3)  does  not  meet  the  x-axis  at  all,  and  that  accordingly  the 
equation  rc^  — 2a;+2=0  has  two  imaginary  roots. 

Writing  the  general  equation  of  the  second  degree  and  one  unknown, 
as  ax''-\-hx-\-c  =0,  in  (1)  h^-^ac>0,  in  (2)  h^-^ac  =0,  in  (3)  h^-4:ac<Q. 
Hence  the  above  graphs  illustrate  the  fact  that  the  character  of  the 
roots  of  an  equation  of  the  form  ax'^-\-hx-\-c=Q  can  be  determined  by  an 
inspection  of  the  value  of  6^  — 4ac  in  the  givene  quation.     (See  Art.  271.) 

421.  Abbreviated  Graphical  Solution  of  a  Quadratic 
of  One  Unknown.  The  graphical  solution  of  an  equa- 
tion like  a;2  +  4a:  — 5  =  0  can  often  be  much  abbreviated  by 
constructing  the  graph  oi  y=  —x^,  and  afterward  that  of  the 
linear  equation  y  =  4:X  — 5  on  the  same  figure ;  and  then  meas- 
uring the  abscissa  of  each  point  of  intersection  of  the  two 
graphs.  For  at  a  point  of  intersection  of  the  two  graphs, 
the  ordinates  of  the  two  graphs  are  identical,  and  hence  the 
values  of  y  in  the  two  equations  are  equal;  that  is,  we  have 
4a;— 5=  — x^,  or  x^  +  4:X—5  =  0. 

In  using  this  method  the  saving  of  labor  arises  from  the 
fact  that  the  graph  oi  y=  —x"^,  having  been  once  con- 
structed, can  be  used  repeatedly  in  solving  a  succession  of 
examples,  and  also  from  the  fact  that  each  straight  line 
used  is  constructed  by  constructing  two  points  only. 

In  general,  to  solve   graphically   an  equation   of  the  form 

ax^  +  hx+c  =  0  by   the   abbreviated  method,   reduce   the  equu- 

b         c 
Hon  to  the  form  x^ +-  x  +  -  =0;    construct    the    graph    of 
a         a 


434  ALGEBRA. 

h  c 

y=—x^    and  also    the   graph    of  y  =-x  +  -  ;  measure  the 

a         a 

abscissa  of  each  point  of  intersection  of  the  two  graphs, 
EXERCISE   148 
Solve  both  graphically  and  algebraically: 

1.  a;2_4  =  o.  7.  Sx^-l=x. 

2.  x^-Sx-4:  =  0.  8.  a;2-4x+l=0. 

3.  4a;2  +  8a:-5  =  0.  9.  5a;2_i8a.+  i6_0. 

4.  x2-6x  +  9  =  0.  10.  x'^-2a:  =  0. 

K    3x  +  5_o_2a;-5  11.  a:^-2a:-l  =  0  (make  the 

x+4  x—2'  algebraic  solution  by  the  fac- 

6.  4x'^  4-  4x  + 1  =  0.  torial  method) . 

12.  How  could  ex.  10  be  solved  graphically  by  using 
y=—x"''  as  an  auxiliary  graph? 

13.  In  general  how  may  the  graphical  solution  of  equa- 
tions of  the  form  ax^^-hx-{-c  =  0  be  faciUtated  by  the  use  of 
2/=  —a:^  as  an  auxihary  graph? 

14.  Solve  graphically  x^  —  2a;  + 1-  =  0. 

15.  Also  3a;^- 6a; -1=0.      *  16.  Also  xHa;^- 2a; -1=0. 
Solve  both  graphically  and  algebraically: 

17.  a;3-x--6a;  =  0.  19.  a;^-5a;2+4  =  0. 

18.  a;^-2a;2-2a;+4  =  0.  20.  7^-'^x^+l  =  (). 

SOME   APPLIED    GRAPHS. 

422.  "Wider  Application  of  G-raphs.  Besides  their  use 
in  ordinary  algebra,  graphs  may  be  used  to  represent  the 
properties  of  a  great  variety  of  functions,  in  particular  those 
occurring  in  the  various  departments  of  science,  and  in 
business  life. 

Sometimes  it  is  found  convenient  to  use  a  different  scale 
in  laying  off  magnitudes  on  one  axis  from  that  used  on  the 
other  axis. 


GRAPHS, 


435 


EXERCISE  149. 

1.  Graph  C  =  -  (F-32)   making  the  scale  on  the  C-axis 

y 

but  one-half  as  large  as  that  on  the  F-axis. 

2.  Graph  V  =  -  and  on  the  graph   obtained  measure  the 

value  of  V  when  P  =  1 .5. 

3.  Construct  the  graph  of  s=lQ.lfi  making  the  s-scale  but 
one-tenth  as  large  as  the  i-scale. 

4.  A  thermometer  reads  as  follows  at  different  hours  dur- 
ing the  day: 


Hour 

7  a.m. 

8  a.m. 

9  a.m. 

10  a.m. 

11  a.m. 

12  a.m. 

1  p.m. 

2  p.m. 

Temperature  .... 

50° 

51° 

54° 

59° 

65° 

71° 

75° 

78° 

Hour 

3  p.m. 

4  p.m. 

5  p.m. 

6  p.m. 

7  p.m. 

8  p.m. 

9  p.m. 

10  p.m. 

Temperature 

78° 

77°  " 

71° 

65° 

60° 

57° 

55° 

51° 

Let  the  pupil  construct  a  graph  showing  the  relation  be- 
tween the  temperature  above  60°  (taken  as  plus)  and  that 
below  (taken  as  minus),  and  the  hour  of  the  day.  Let  him 
then  point  out  some  facts  to  be  learned  from  this  graph. 

5.  The  average  temperature  on  the  first  day  in  each  month 
for  the  last  thirty  years  in  New  York  City  has  been  as 
follows : 

New  York. 


Date 

Jan.  1. 

Feb.  1. 

March  1. 

April  1. 

Mayl. 

June  1. 

Temperature 

31° 

31° 

35° 

42° 

54° 

64° 

Date 

Julyl. 

Aug.  1. 

Sept.  1. 

Oct.  1. 

Nov.  1. 

Dec.  1. 

Temperature 

71° 

73° 

69° 

61° 

49° 

39' 

Let  the  pupil  graph  these  data. 

The  corresponding  temperatures  in  London  were  as  follows: 


436 


ALGEBRA. 


London. 


Date 

Jan. 1. 

Feb.  1. 

March  1. 

April  1. 

May!. 

June  1. 

Temperature 

37° 

SS" 

40° 

45° 

50° 

57° 

Date 

July  1.    1   Aug.  1. 

Sept.  1. 

Oct.  1. 

Nov.  1. 

Dec.  1. 

Temperature 

62» 

62» 

59" 

54» 

46° 

41^ 

Let  the  pupil  graph  these  results  on  the  same  paper  with 
the  graph  of  the  New  York  temperatures  and  then  compare 
the  two  curves  of  annual  temperature/ and  give  three  facts 
which  may  be  inferred  from  these  curves. 

6.  The  following  table  shows  the  number  of  years  which  a 
person  having  attained  a  certain  age  may  expect  to  live. 
Let  the  pupil  construct  a  graph  of  life  expectancy  from  the 
data: 


Ace  in  Years 

0 

2 

4 

6 

8 

10 

20 

30 

Life  Expectancy  in  Years 

38.7 

47.6 

50.8 

51.2 

50.2 

48.8 

41.5 

34.3 

Age  in  Years 

I« 

50 

60 

70 

80 

90 

100 

Lif«^  F/Xppctan'^y  in  Yparj* 

27.6 

21.1 

14.3 

9.2 

5.2 

3.2 

2.3 

From  this  graph  let  each  pupil  determine  his  or  her  life 
expectancy  at  the  present  time,  and  also  that  of  a  number  of 
acquaintances  of  various  ages. 

7.  Graph  2/ =  log^x.  8.  Graph  2/  =  logioX. 

9.  Graph  y  =  x^  (logarithms  may  be  used  to  advantage  in 
part  of  the  work). 

10.  Graph2/  =  a;3.  11.  Graph  2/ =  a:  -  a. 

12.  Construct  the  parallelogram  whose  sides  are  the  graphs 
of  the  equations  Sy— 4a;— 13  =  0,  Sy—4iX+19  =  0,  y  =  S, 
2/  =  —  1 ;  find  the  co-ordinates  of  the  vertices  of  this  paral- 
lelogram, and  also  its  area. 


ANSWERS, 


Exercise  1. 

18.  5a  +  lb. 

17.  ^^  +  ^y      9a:  +  ^. 

14.  6a2  = 

=  2(a- 

6). 

32                     3^3 

15.  4(a2 

-96)  < 

(7a  4 

•^')'-            18.5.3^          6^^     .-. 

5(a3  +  6«) 

16.  {x  - 

10^2)  (a;3  ^  yz^  ^  2ax\      "  '"        {x  ~  ly^f  ' 

(a;  +  2^)» 

Exercise  2. 

1.  10. 

12. 

3.                          30.  18. 

44.  -V-. 

2.  1^. 

13. 

6.                           31.  11. 

45.  i 

3.  30. 

16. 

18.                        32.  3. 

46.  -W-. 

4.  15. 

17. 

12.      ,                  33.  f. 

47.  3. 

5.  9. 

19. 

16.                •  ■      34.  f . 

48.  2. 

6.  25. 

24. 

6.                          35.  1. 

49.  6. 

7.  6. 

25. 

108.                       36.  f. 

58.  13. 

11.  7. 

29. 

63.                        42.  15. 
Exercise  3. 

59.  6. 

1.  -5. 

7.  Ix^.                             17.  a  +  6 

2.  -6. 

11.  4aa;.                           18.  Ix"  - 

\\y\ 

3.  2a;. 

14.  2a;.                             19.  26^1 

20.  7x2  ^ 

■2y\ 

26.  -  2a;  -  4^  +  1z. 

'24.  9a;  +  12a:2;. 

i»»  . 

27.  -  xy  +  2aa;  +  y"^  - 

-Zx\ 

25.  2a;V 

+  4a:3/^ 

28.  -  ^x-y  -  20. 
Exercise  4. 

1.  4a6.  15.  -  1  -  2a;  +  2a;2  +  a:»  +  3JC*. 

2.  -  4a;.  16.  llxy"  -  x'y''  -  9a;V. 

3.  -  X.  22.  4a;"  -  2a:'«  -  a;^  +  2. 

4.  8a;.  23.  4a;3  -  2a;  -  2. 

7.  x^  -  5a;.  24.  2ar«  +  63?^  -  2a;  -  4. 

i 


ii 


1.  5a   -  6. 

2.  a;  +  1. 

3.  1  -  a;. 

4.  -1. 

5.  2a;  +  1. 

6.  -  X  +  3y. 

7.  1  -  2a;. 

8.  9a;  -  1. 


ANSWERS, 

FiXercise  5. 

t 

9.  4. 

17.  2c  -  ft  -  d 

10.  4a;  -  1. 

18.  3a;  -  2ar». 

11.  0. 

19.  -  73^  +  x^  -  2x  - 

-1. 

12.  a  -  1. 

20.  -  X. 

13.  a 

21.  2. 

14.  0. 

22.  -  2y. 

i5rx-+ 1. 

23.  -  3a;. 

16.  6. 

Exercise  7. 

6.  1  -  5ab  +  2bc-  d  +  x. 

6.  d^  -  a  +  2a;  +  10^/. 

7.  -  7a6  -  2c  +  c^-H  4a;  -  2V^, 

8.  41/^  -  5a;  +  a;2  +  a;3  +  a;*  -  4. 


1.  2a;*  -  3a;'  -  5a;  -  2. 

2.  4a;^0'^  +  33:^^2:  —  x^^z. 

3.  31/3  -  31/2  -  1. 

4.  3(a;  +  y)  +  {y  +  z). 

9.  12.  11.  3;  13.  13.  12a;. 

10.  2 ;  f .  12.  12a;  -  3.  14.  2a;2  -  ^z\ 

15.  (I  +  a  -  2c)a;5  -  (3  +  c)x''  -  (1  +  a  -  3c)ar»  -  (2a  +  b)x^  +  2. 

17.  3a'  -  10a6  +  ^aW.  .     21.  4a;3  -  2x^y  +  2xy^  +  7^'. 

18.  First,  ar*  -  a:2  +  a;  -  1.  22.  -  2a;'  +  2a;^2  ^  3^3^ 

19.  First,  -  a;'  +  2a;2  -  3a;  +  1.  23.  6a;'  -  2a;2^  -  3^'. 

20.  2a;'  -  2a;V  +  6a;^2  +  5^3  24.  10a;'  -  ^x'y  +  4a;^^  -  y^. 


5.  -  Sa;*. 

6.  15a;2. 

7.  -12a2a;2. 

8.  42a;V- 

9.  -  2la?ocy. 
10.  20a}bcd\ 
14.  -  12a;2« 


1.  2a;2  -  7ar  -  4. 

2.  3a;2  -  7a;  -  6. 

3.  2a;'*  -  9a;  -  35. 

4.  12a;'  -  25a;;?/  +  12^'. 

5.  28a;*  +  x'^y^  -  15y*. 

6.  30a;2^'*  +  xy  -  42. 


Exercise  8. 

15.  -  35a-2V. 

16.  -  3a;"  +  V" 

18.  6a^x  +  9axl 

19.  -  15iv  +  10^^'- 
25.  2a;"  +  '  -  3a;"  +  ^. 

27.  63;'"^*"  -  15a;*"^  +  *. 
29.  20a;  -  12a;^ 

Exercise  9. 

7.  32a5c  -  2a6*c». 

8.  SSx^y  +  xY  - '^^y^* 

9.  a'  +  6'. 

10.  X*  -  y\ 

11.  8a:*-  2a:'  +  a;2-  1. 

12.  6a.''  -  19a;=*^  +  21xy^  -  IQ/, 


ANSWERS.  iii 


13.  23*  -  5x*  -  23r^  t  9x^  -  7a;  +  3. 

14.  3x^2/  -  ^Ox^y^  +  4a:y  +  g^,^  ^  ys 

15.  a;^  -  oa:V  +  lOx^  -  10a:V  +  ^^P*  -  y*. 

16.  4a:6  +  g^i  _  jg^  ^  22^2  -  21a:  +  6. 

17.  a:6  -  a;6  _  7^  _^  33JJ  +  17^2  -  5a;  -  20. 

18.  a:«  -  6a;V  +  Ga:^'  -  y^-  20.  a*  +  a^ft^  +  ft* 

19.  a;*  -  14a;3  +  49a;2  -  4.  21.  16a;*  +  S6xY  +  81^. 

22.  x''  -  9x^y^  +  7a; V  +  ISar*^*  -  lOa;^^^  4  g^^a  _  y^^ 

23.  -  a.\+  2aa;*  +  Sa^a^s  _  lea^a;^  -  16a*a;  +  32a*. 

24.  a^  +  b^  +  XT'  +  Sab''  +  Zo?b. 


25.  aW  +  c^d^ 

- 

a'c'  -  b^d\ 

26.  a;"  +  1  -  a;" 

-1 

-6a;'»-2- 2a;H-4. 

27.  a;2"  +  1  ^  a;^' 

I 

2a;2»  -  1  4  3^2«  -  2  _  joa:2n  -  3^ 

28.  a;~  -  *  +  a;«  ■ 

-3 

-  a;«  - '  +  a;"  -  ^  -  a;"  +  7a;«  +  1  +  10a;"  +  » 
Exercise  10. 

1.  -ar. 

3.  x\                               5.  X-  2a;»  -  2a?^ 

2.  2a;2  -  8. 

4.  -  20a?.                         6.  2  -  x\ 

7.  3a;2  -  6a;, 

13.  a;2  +  10a;  -  16. 

8.  -  2a6  -  861 

14.  0. 

9.  a"  -  462  +  126c 

- 

9c2.                 15.  a;2  -  5a;  +  8. 

10.  y^  -  2yz  +  z\ 

16_^3a;2-10a;y  +  2/'. 

11.  2a;  +  1. 

17.  x^  -z\ 

12.  2a;  +  2a;^  '; 

18.  4a^  —  ax  +  bx  +  my  +  cy. 

19.  0. 

22. 

40.                     25.  7.                      28.  -  11. 

20.  4a;2. 

23. 

-  3.                   26.  -  2.                   29.  1. 

21.  -  12. 

24. 

-  2.                   27.  -  1.                   30.  5. 

31.  29. 

Exercise  11. 

1.  -3. 

6.  -  7y^ 

2.  -  3a;». 

17.  x^-x  +  1. 

3.  -  2a. 

18.  3a;2  -  7a;  +  1. 

4.  hxy. 

23.  2a;  -  5a;2  +  3a;2  -  » 

5.  -  X.            V- 

24.  x*-2x^  +  3a;2  +  x. 
Exercise  12. 

1.  3a;  +  1. 

3.  4a;  -  5y.                      5.  3a;  +  7. 

2.  2a;  +  1. 

4.  3a;  +  iy,                     Q,  3x  -  5^. 

iv  ANSWERS. 

7.  3a  +  4c.  16.  a;^  -  3a;  +  1. 

8.  -5a: +  8.  17.  Ix^  +  ?>x  +  l.> 

9.  4a;  +  ^.  18.  '6a^  -  Aax  +  x"^. 

10.  a  +  26.  19.  2y'  -  Ay-"  +  y  -  1. 

11.  a;'*  +  a;y  +  y\  20.  c*  +  c^a;^  +  a;*. 

12.  9a;2  -  6a;  +  4.  21.  23^  -  Zx"^  +  4a;  -  5. 

13.  3a;  -  7.  22.  20?  -  x  ^  \. 

14.  25  +  20a;  +  16a;2.  23.  3a;^  +  4a;2^  +  bxy"^  +  2^. 

15.  ^a^x"  -  2axy''  +  y^.  24.  2a;*  -  3a;2y  -  2^^ 

25.  a;3  +  2x'V  +  4a;y2  +  8^^ 

26.  a;*  -  2a;V  +  4a; V  -  8a;^^  +  16^*. 

27.  x^  —  a;*y  +  a;^^^  -  a;^  +  a;?/*  —  if". 

28.  64.^^  +  16a;*^2  +  4^2^*  +  ^e^ 

29.  ^x"  -  5a;  -  1.  87.  2a;*  -  4a;3  +  Zx^  -  2a;  +  1. 

30.  3a;2  -  a;  -  5.  38.  2xy  -  2xz  -  Zyz, 

31.  2a;3  -  4a;2  -  a;  +  3.  39.  x"-  -  3a;  +  1. 

32.  2a;'  +  3a;2^  -  4a;^2  4.  ^3^  4Q   ^^z  _  3^2  +  a;  -  5. 

33.  3a3  -  ^a^h  +  3a62  _  26'.  41.  2a;"  -  3a;"  -  \ 

34.  x''  +  I/'''  -  z^  -  xz  ^  xy  -  yz.  42.  4ar»«  +  Zx^^  -  a;". 

35.  c^  -I-  d?  -vn?  -  cd  -  en  -  dn.  43.  4a;"  +  ^  -  3a;"  +  a;"  - ». 

36.  2/*  +  2i/2  +  3^2  ^.  2i^  4-  1.  44.  3a;"  -  1  +  2a;"  -  2  -  3a;"  -  \ 

Exercise  13. 

1.  |a;3  -  -V^  +  |.  3.  2a;*  -  t|a;3  _  o^,  _  1. 

2.  |a;3  -  |a;2  +  f.  4.  2.88a;3  +  10.86a;  -  19.2. 

5.  5.4a;*  -  3.3a;3  +  10.  la;^  +  1.32a;  -  .08. 

6.  6.75a;*  +  \.2^y  +  15.84a;V  +  13.443/*. 

7.  fa;2  -  4a;  +  |.  10.  1.8a;2  -  3.2a;  +  0.48. 

8.  ^x"  +  fa;  -  1.  11.  0.5a;2  -  1.8a;  +  3.5. 

9.  |a;2  +  ^xy  +  f^/^  12.  3a;2  +  4.8a;?/  -  21.5^^ 


Exercise  14. 

1.  5. 

8.  -  i 

15.  -  |. 

22.  3. 

2.  -3. 

9.  0. 

16.  -  f. 

23.  1. 

3.  -2. 

10.  6. 

17.  -  f. 

24.  1. 

4.  2. 

11.  -  5. 

18.  -  f. 

25.  6. 

6.  -  |. 

12.  11. 

19.  -  h 

26.  14. 

6.  -  -^^. 

13.  0. 

20.  -  f . 

27.  -Y. 

7.  ^,  14.  -  2.  21.  f .  28.  -  3. 


ANSWEBS. 


Exercise  W. 


1.  12  and  36  marbles. 

2.  A,  $61 ;  B,  $39. 

3.  John,  36  ;  William,  60  ex. 

4.  1st,  $26 ;  2d,  $37 ;  3d,  $35. 
6.  80  miles. 

6.  27,  28,  29. 

7.  19,  73. 


8.  Ist,  22;  2d,  11;  3d,  17. 

9.  Horse,  $67  ;  cow,  $27. 

10.  A,  $97  ;  B,  $194 ;  C,  $194. 

11.  14,  21. 

12.  13,  14,  15,  16,  17. 

13.  9,  15. 

14.  1st,  $280;  2d,  $140;  3d,  $80. 


15.  Daughter,  $960 ;  sons,  each,  $1770. 

16.  Father,  48  ;  son,  12  years. 

17.  Father,  52  ;  son,  31 ;  daughter,  26  years. 

18.  Father,  84 ;  son,  42  years.  20.  19,  21,  23. 

19.  11,  25.  21.  9,  14. 

22  Eldest,  $21 ;  rest,  in  order,  $13,  $9,  $7,  and  $6. 


23.  21,  54. 

24.  21,  22. 

25.  14,  16. 

26.  Father,  44  ;  son,  20  years. 

27.  4 1  hours ;  36  miles. 

28.  8  miles. 

29.  36  miles. 

30.  38  days. 

31..  A,  25  ;  B,  15  years. 

32.  Father,  55 ;  son,  25  yeais. 

33.  Silk,  $3.30 ;  cloth,  $1.10. 

34.  9. 


35.  15  of  each. 

36.  17  of  each. 

37.  7  bills  ;  14  quarters. 

38.  17  halves  ;  3  dimes. 

39.  Son,  $375 ;  daughter,  $15(> 

40.  11  beggars. 

41.  A,  $42;  B,  $37;   C,  $33. 

42.  5  @  32  cts. ;  7  @  20  cts. 

43.  48  feet. 

44.  A,  $21  ;  B,  $68. 

45.  72  pounds. 

46.  A,  24  miles. 


I^y. 


Exercise  17. 

6.  0.3a:3  -  0.7a:V  +  3.3a:2/» 

7.  ^x^  -  Ix  +  S. 

8.  3.6a;2  -f  1.29.ry  -  0.6^'. 

9.  fx*  +  la'^x'  +  fa*. 
10.  ^x*  -  Ux'  +  ilx  -  1. 

11.  4.8a'  +  4.55a26  +  2.05a62  +  1361 

12.  0.22r»  -  l.bx'^i/  +  l.Sx^'  +  4.2^'. 

13.  1.82;V  -  2.73:cV  -  l-^^^^  +  2.7a;V. 

14.  4.S2^  -  17.95^2^  +  18.45ar?/2  -  6.3^'^ 

15.  2x2  _  4a;y  +  2^2 

IQ,  z^  -  2x^y  +  2a;y  +  3xY  -  ^^y*  -  8y*. 


i^iSf. 


2. 

^x^- 

-h 

3. 

4x'- 

-  Z.by\ 

4. 

2.6  +  0.92r  - 

-a:». 

6. 

\2^- 

-  i:r='  - 

\x-V 

Vi  ANSWEES. 

17.  2  -  3x  +  3x^  -  3x^  +  Sx*- 

18.  1  -2x  +  2x^  -2x*  +  2x^ 

19.  a:*  -  a;'  +  2a:2  -  3a;  +  5 

20.  1  +  2a;  +  7a;2  +  20a;3  +  61a:* 

21.  |a;2  -  2a;^  +  iy\  26.  23^  -  2.5a:'-'^  -  O.Sxy^  +  1. 

22.  far»  -  |a;V  +  h^y^  27.  2z  -  x. 

23.  6x-  ^y  -  |.  28.  0. 

24.  1.6a:='  -  2xy  +  2Ay\  29.  3a;  +  7^. 

25.  3.5a;2  -  3a;  +  1.5.  30.  2a  -  12c  +  84(Z. 

31.  3  +  4a;  -  2^/.  34.  -  3.  37.  -  1. 

32.  5d'b  +  46'.  35.  -  1.  38.  x^  +  a;  -  1. 

33.  2.  36.  -  8.  39.  x^  -  3. 

40.  38. 


Exercise  18. 

1.  n'  +  2ny  +  y^. 

11.  x^  -  z\ 

2.  c^  -  2ca;  +  a;'. 

12.  ^2  _  9. 

3.  \x^  -  \xy  +  3/2. 

14.  49a;'^  -  16^'. 

4.  9a;2  -  12a;2/  +  ^y". 

18.  4a;2«  -  25^2«, 

Exercise  19. 

1.  d}  +  2ah  +  62  _  9.  9.  a;*  +  a^y  ^.  ^^i^ 

3.  15  —  2x  —  x^.  11.  4a;*  -  29a;2  +  25. 

5.  4a:2  -  9^^  _  g^/  -  1.  16.  a^  +  2a6  +  6^  -  c'  +  2c  — 1. 

6.  a;*  +  6a;3  +  9a;2  -  4.  17.  a;*  +  2/*  -  a;*^  -  1. 

Exercise  20. 

1.  4x2  +  ^2  ^  1  ^  4^^  +  4a-  +  2^/. 

2.  a;'^  +  4^2  4.  422  _  /^xy  +  4a;2;  -  ^yz. 

3.  9a;2  +  4^2  +  25  -  12a;^  -  30a;  +  20^. 

4.  4a2  +  62  +  9c2  -  4a6  +  12ac  -  66c. 


9.  x'  +  2/2  - 

f02  +   l 

-2a;^ 

+  2a;2;  ~  2a;  -  2yz  +  2y  -  tz. 

20.  a*  +  46* 

+  4a6- 

-1. 

Exercise  21. 

1.  a;»  +  7a;  +  10. 

5.  a;2  -  6a;  -  7. 

2.  a;2  -  8a;  +  15. 

20.  a2  +  (6  -  l)a  -  6. 

3.  a;2  -  3a;  -  28. 

23.  a;2  +  4a;2/  +  4^2  _  53.  _  \^-\^ 

4.  ic'  +  4a;  -  32, 

27,  a2  +  2a6  +  62  +  a  +  6  -  12. 

ANSWERS.  yii 

Exercise  22. 

1.  2a;'  +  7a;  +  6.  3.  Sx^  -  7a;  +  2. 

2.  2a;»  +  a;  -  10.  4.  5x^  -  4x  -  U 

Exercise  23. 

1.  a  +  X.  9.  a  +  6  -  2c. 

2.  3  +  2a;.  10.  2x^  -  y""  -  I. 

Exercise  24. 

1.  a}  -2a  +  4.  7.  x*  -  a;^^^  4.  ^4^ 

2.  a;2  +  a;  +  1.  8.  a^  _  2gt  +  1  +  aa;  -  a;  +  a;*. 

3.  9a;2  +  12a;  +  16.  9.  c^  -  c  +  ca;  +  1  -  2a;  +  x\ 

4.  1  -  2a;"^  +  4a;*.  10.  4  +  2x  +  2y  +  x"^  +  2xy  +  y'. 

5.  25  +  5ar'  +  a*.  11.  a;*^*  +  x^y^  -  2xy  +  1. 

6.  9a*  -  3aV*  +  2/®-  .     12.  9a;*  -  15a;V  +  25^«. 

Exercise  25. 

1.  a'  +  Sab  +  96^  12.  3a;  +  5^' and  3a;  -  5y\ 

2.  a»  +  2a26  +  4ab^  +  86».  13.  a;^  +  9,  a;^  -  9,  a;  +  3,  a;  -  3. 

3.  a;*  -  ar«  +  a;2  -  a;  +  1.  14.  a;  -  2. 

4.  a;*  +  33^"  +  9a;2  +  27a;  +  8;.  15.  1  +  x. 
11.  1  +  4a;.                                          16.  a;«—  Sy', 

17.  x^  +  ^,  7^  —  y'^,  ^  -^  y,  ^  -  y- 

21.  a;«  +  8^^   a;*  -  81/',   a;*  -  4.y2,   x^  +  2^,   a;^  -  2y. 
23.  Ist,  all  integral  values ;  2d,  all  even  integers. 

Exercise  26. 

1.  a;«(2a;  +  5).  3.  a:(a;  +  1).  8.  a;'(l  -x-  x^). 

2.  xix"  -  2).  6.  7a(l  +  la").  15.  arb^&\l  +  lie). 

Exercise  27. 

1.  (2a;  +  yy.  3.  (5a;  -  1)^.  6.  c(7  +  2bc)\ 

2.  (4a  -  Zy)\  4.  (a;  -  l()y)\  17.  {a  -  b  -  g)\ 

Exercise  28. 

1.  {x  +  3)  (a;  -  3).  21.  (15a;«  +  y)  (15a;«  -  y). 

2.  (5  +  4a)  (5  -  4a).  24.  (a;  +  y  +  1)  (a;  +  y  -  1). 

3.  (2a  +  76)  (2a  -  76).  25.  {x  +  y  +  1)  {x  -  y  -  1). 

4.  3(a;  +  2y)  {x  -  2y).  29.  (4a;  +  2^+1)  {2y  -  2a;  -  1). 
6.  (10  +  9m)  (10  -  9m).  30.  (11a  -  86)  (9a  -  26). 

9.  xix""  +  1)  (a;  +  1)  {x  -  1).  31.  y(a;«^*+2;8)(ar»3/H2;*)(a;3y-2?*). 


viii  ANSWERS, 

Exercise  29. 

1.  (5a;  -  8^)  (8y  -  Sx).  6.  {a  +  ^  +  x)  [a  +  ^  -  x). 

2.  (3a  -36  +  5)  (3a  -  36  -  5).        7.  (a*  +  x^  +  y)  {a'  -  x^  -  yi 

3.  (a  -  6  +  1)  (a  -  6  -  1).  8.  {x  +  y  + 1)  (x  -  y  -  1). 

4.  (3a;  +  2y  +  z)  {3x  +  2y  -  z).         9.  {1  +  x  ~  2/)  {1  -  x  -¥  y). 
6.  {x  —  a  -^^  y){x  —  a~  y).  10.  (c  +  a  -  6)  (c  —  a  +  6^ 

Exercise  30. 

1.  {c^  -V  ex  -\r  x^)  (c2  -  ex  +  x"^). 

2.  [x^  +  X  +  1)  {x'  -x  +  1). 

3.  (2a:2  +  3a;  -  1)  (2a;2  -  3a;  -  1). 

4.  (2a2  -  3a6  -  36^)  (2a2  +  3a6  -  36^). 
6.  (3a;=*  +  3a:^  +  2y'')  (3a;'  -  3a;^  +  2y^). 

13.  (a^  +  2a6  +  26'^)  {a"  -  2ab  +  26'^). 

Exercise  31. 

1.  (1  -  2a;)2.  4.  a;(l  +  xf  (1  -  x)\ 

2.  Zx{2y  +  x)  {2y  -  x).  9.  2a;(4a;2  +  3a;  +  1)  (ix-Sx  r  1). 

3.  (3  -  a:  +  ^)  (3  -  a;  -  y).  12.  (a;^  -  2a;  -  1)  (a;  -  l)^. 

17.  {x-\-y  +  l){x  +y-l){l-^-  x-y){l-x+  y). 

Exercise  32. 

1.  {x  +  8)  (a;  +  2).  31.  x{x  +  4)  (a;  +  3)  [x  -  4)  (a;-  3). 

2.  {x  +  2)  {x  -  3).  32.  Seven  factors. 

3.  {x  +  3)  (a;  -  2).  37.  {x  +  a)  {x  +  6). 

4.  {x  +  11)  {x  -  4}.  38.  {x  +  2a)  {x-  36). 

5.  {x  -  5)  (a;  — 6).  39.  (a;  -  a)  (a;  -  26^).    . 

12.  {x"  +  4)  {x  +  3)  (a;  -  3).  43.  2(a;  +  1)^  {x  +  4)  (a;  -  2). 
24.  {x'  -  8)  (a;  +  1)  {x  -  1). 

Exercise  33. 

1.  (2a;  +  1)  {X  +  1).  5.  (3a;  -  5)  (2a;  +  1). 

2.  (3a:  -  2)  (a;  -  4).  6.  (a;  +  3)  (2a;  -  1). 

3.  (2a;  +  1)  (a;  +  2).  7.  2a;(a;  +  4)  (3a;  -  2). 

4.  (3a;  +  1)  (a;  -f  3).  20.  (2a;  +  3)(a;  +  1)  (2a;-.3)  (a;-l). 

21.  (3a;  +  2)  (a;  +  4)  (3a;  -  2)  (a;  -  4). 

24.  {o?  +  62 j  (5a  +  46)  (5a  -  46). 

27.  (5a  +  46)  (a  +  6)  (5a  -  46)  (a  -  6). 

31.  (a  +  6  +  8)  (a  +  6  -  3). 

32.  (3a;  -  Zy  -  2z)  {x  -  y  ^  Zz). 


ANSWERS. 


bL 


33.  (3a:'  +  6x  +  A)  {x  +  3)  {x  -  1).     35.  2(1  +  3a:)  (2  -  x), 

34.  4x{x  +  4){x  +  2)  {x  + 1)  (a;- 1). 

Exercise  34. 

1.  {m-n)  (m'  +  wn  +  w').  9-  x{3x-j-a)  (dx^  —  Sax  +  a^). 

2.  {c+2d)  {c''  —  2cd-i-Ad').         10.  (8a;  — r)  {Gix"  +  8xy' +  y*) , 

3.  (3  — a;)  {9  +  3x-\-x^).  11.  a(l  +  7a)  (1— 7a  +  49a'). 

12.  {a-{-x)  (a  —  x)  {a^ — ax-^x'^)  {a'  -\- ax -{-  x'^). 

13.  {x'-{-y)  {x^-y)  {x'-x'y -j-y^)  (x'+x'y  +  y^). 

17.  (a -f  &  + 1)  (a^ +2a&  +  &2  _  a  _  &  +  1). 

18.  (5  +  2&  — a)  (25  — 10&  +  5a  +  4&2— 4a&  +  an. 

19.  (2  — c  — d)  (4  +  2c  +  2(2-f  c2  +  2cd+d2). 
22.  {x -\- y)  {x*  —  x^y-\-xhj^ — xy^-\-y*). 


18.  As  tha  24th.    29.  As  the  23d. 

34.  As  the  12th.     35.  As  the  22d, 

Exercise  35. 

1.  (a  +  6)  (a:  -F  ^). 

18.  {X  -  y)  {2x  +  2y-  1). 

2.  (a;  -  a){x  +  c)w 

20.  {x  -  1)  (a;2  +  3a:  +  3). 

3.  (5^  -  3)  (x  -  2). 

23.  (3a  -  x)  (3a  +  2a:)  (a  -  a;). 

4.  {m-2y)  (3a  ^  4n). 

' 

24.  {x  -  2)  (a:  +  3)  {x  -  1). 

6.  a;(a  +  3)  (a  +  c). 

25.  {x  +  3)  (2a:  -  5)  (2a:  -  1). 

6.  2/(3a  -  5n)  (a  +  6). 

26.  (2a:  +  1)  (4a:  -  3)  (a;  +  1). 

7.  a:(a:2  +  2),  (a;  +  1). 

27.  (2a:  -  3)  (4a:  -  3)  (a:  -  2). 

8.  2a:(a:  +  a)  (a:^  a)  {x 

-1). 

28.  {x  +  2)  (a:  +  1)  [x  -  3). 

9.  [y'  +  1)  (y  +  1). 

29.  {x  +  3)  {x  -  2)  [x  -  4). 

16.  {X  +  4)  (a:  +  2)\ 

30.  {x  -  2)  (a:  -  1)  {x  -  5). 

17.  (a  +  3)  (a^  -  3). 

31.  {X  -  3)  (2a;  -  1)  (3a;  -  l)c 

Exercise  36. 

10.  3a:(a:2  +  a:  +  1)  (a;^  -  a:  +  1)  (a;  +  1)  (a;  -  1). 

11.  (2a  +  1)  (a  +  1)  (2a  -  1)  (a  -  1). 

12.  2(a:2  +  2a;  +  2)  (a:^  -  2a;  +  2)  [x'  +  2)  {a^  -  2). 
15.  5a(a:«  +  a;3  +  1)  (a;*  +'a:  +  1)  (a;  -  1). 

23.  {x  +  2)  (a;  -  1)  {x^  -  a:  +  2). 

34.  (a'  +  5)  (a  +  2)  (a  -  2). 

35.  (G  +  d  -  1)  {c^  +  2cd  +  (P  +  c  +  d  +  1). 

36.  {x  -  y)  {x-y-\-  2). 

39.  (a  +  3)  (a  +  2)  (a  -  3)  (a  -  2). 

41.  (a  +  6  +  c)  (a  +  6  -  c)  (a  -  6  +  c)  {a  -  6  -  e), 

43.  (2  +  n)  (16  -  8n  +  Ari^  -  2n'  +  n*). 

46.  ia(2a;  +  y)  [ix^  -  2xy  +  y^). 


ANSWERS. 

47.  (1  +  x')  (1  +  xy  (1  -  x). 

48.  &{x  -  3)  (4  -  x). 

51.  {x  +  1)  (a;  +  2)  {x  -  3). 
63.  (a:*  +  62;^  +  ^*)  (a?  +  yf  {x  -  y)\ 
54.  (a:2  +  ^2  ^  ^^^  (3,  +  2;)  (a;  -  z). 
66.  (a  +  6)  (a  -  76). 

61.  {X  -  ly  {x  +  2)2. 

62.  (a  +  62)  \a  -  6^)  (i  _  a;)  (i  +  a;  +  a:'). 
66.  (1  +  2a6c  -  Zxyz)  (1  -  2a6c  +  Zxyz). 

67.  {dbc  -  772W/))  {ax  -  my).  71.  (2a;  +  5)  (2a;  -  3)  (a;  -  1). 

69.  {x  -  2)  (a;  +  5)  (2a;  +  1).  72.  {x  +  1)  (a;  +  2)  (3a;  -  2). 


Exercise  37. 

1.  2a6. 

7.  17aa;3.                 13.  a  -  a;. 

19.  4aa;(a  -  x) 

2.  5a;V. 

8.  aVy.                  14.  a;  +  1. 

20.  x{x  -  1). 

3.  a6c2. 

9.  a  +  6.                 15.  a;  -.1. 

21.  2a;  -  y. 

4.  8a2a;2. 

10.  x-y.                 16.  a(2a  +  1). 

22.  a;(a;  -  2). 

5.  14m2. 

11.  a;  -  3.                 17.  a;  +  1. 

23.  X'  +  9. 

6.  12a;. 

12.  a;(2a;  +  3).           18.  a;  -  3. 

24.  6(1  -  a?). 

25. 

1  +  a  +  a2.                                 26.  a;  -  1. 

Exercise  38. 

1.  a;  +  1. 

6.  3a;  +  4.                11.  a;^  +  4. 

16.  a;(3a;  -  4). 

2.  2a;  -  3. 

7.  2a;  -  1.               12.  a;  +  3. 

17.  a;  -  1. 

3.  a;  -  1. 

8.  a;  +  3.                  13.  a;^  +  a;  +  1. 

18.  a;  -  2. 

4.  3(a;-  1)». 

9.  a;2  +  a;  -  1.         14.  2a^(2a;  -  3). 

19.  a;  -  3. 

5.  a;(2a;  +  1). 

10.  a;2  -  2a;  +  3.       15.  3a;  -  y. 
Exercise  39. 

20.  2a:(a;  -  1). 

1.  6a262. 

5.  12a6c.                         9. 

42a262. 

2.  24a;»^3^. 

6.  I'ZdWcK                    10. 

24a;V. 

3.  36a2a;2^2. 

7.  ^^y?y^z\                     11. 

2a;(a;2  _  jj^ 

4.  48a;3^^ 

8.  48a26-^.                       12. 

ah{a  +  6). 

13.  14a;2(a;  -  3).  20.  a«  -  6«. 

14.  {7?  -  1)  (a;  +  1).  21.  6a:2(a;  +  1)  (a;  -  1). 

15.  {x'  -  y')  {x  -  2y).  22.  3a6(a  +  6)  (a  -  6). 

16.  6a;(a;  +  1)  (a;  -  1)^.  23.  (2a;  +  1)  (a;  +  1)  (2a;  -  1). 

17.  abx'y{x  +  y)  {x  -  yf.  24.  6a;(a;'  -  1)  (a;  -  1). 

18.  {x  +  5)  (a;  -  8)  {x  -  1).  25.  6a;(3a;  +  10)  (2a;  -  7)  (a;  -  3). 

19.  [x  +  2)  (2a;  +  3)  (3a;  -  4).  26.  2a;(l  +  a;'^)  (1  +  a;)  (1  -  ;|7), 


ANSWEBS,  2 

27.  lAx^y^ix  +  ly  {x  -  ly. 

28.  Qx'iSx  4  1)  (a;  -  1)  (3a;  -  1)1 

29.  3Ga;«(2a;  +  3)'  (2a;  -  S)^. 

30.  {x  +  a-¥  l){x-  a-  l){x  +  a-  1). 

31.  {X  -  ly  {x  +  1)2  {x  +  sy  {x  -  3). 

Exercise  40» 

1.  (a;  +  2)  {x^  -  2a;  +  3)  {x^  -  2x  -  1). 

2.  (3a;  +  4)  (a;'-*  -  a;  +  1)  (a;^  +  a;  -  1). 

3.  3a;(2a;  +  1)  (a;^  -  a;  -  1)  {x^  +  x  +  1). 

4.  2(2a;  -  1)  (3a;2  -  5)  (2a;'^  +  5). 

5.  (2a;  +  1)^  {x  -  S)  {x  -  1)  (2a:  -  1). 

6.  (4a;2  -  9)  (9a;'''  -  4). 

'     7.  (a;  -  1)  (3a;2  +  a;  +  1)  (4a;2  +  4a;  -  1). 

8.  {x  +  2)  i^x^  -7x  +  5)  (2a;2  —  x  +1). 

9.  {X  +  3)  (a;  +  6)  (x  -  1)  {x  -  2)  (a;  -  5). 

10.  (X  +  1)  (a;  +  2)  (2a;  -  1)  (3a;  -  2)  (2a;  -  3). 

11.  H.  C.  R,  a'b^ ;  L.  C.  M.,  HOa'S^c^d'. 

12.  H.  C.  F.,  3(a;  +  1)^ ;  L.  C.  M.,  ISa'd^x-"  -  ly  (3a;  +  2)^  (5a;  -  2)». 

13.  H.  C.  F.,   a;2  +  a;  -  2 ;    L.  Q.  M.,    (a;^  +  a;  -  2)  (a;"  -  a;  +  2)  (a;^  + 
+  2). 

14.  H.  C.  F.,  2a;  +  1  ;  L.  C.  M.,  (2a;  +  1)''  (a;  -  1)^  (3a;'''  -  2a;  +  1). 

15.  H.  C.  F.,  a;  +  2 ;  L.  C.  M.,  4(a;  +  2)  (3a;  -  1)^  (a;  -  1)^. 


Exercise  41. 


1    2a. 

3a;* 

2.1?. 


dy 

X 


2 -Sax 

Sxz 

4y^ 

1 

2(a;  -  y) 

a  +  b 
2{a  -  b) 
2 
2a  -  1  3a;  -  4y 

6.  A.  13.        1       - 


10. 
11. 
12. 


x~y. 

x->r  y 

2(x  +  1) 

3 

5 

7. 


2a  2a;  +  3y 

1,  14.  7a;-H8y. 

a  2x^  x+  a 


15. 

1 

x-y 

16. 

x  +  2y  , 

2x  +  Zy 

17. 

2x  +  y, 
2x-y 

18. 

a  +  b  —  c 

a-  b  -  c 

19. 

1  +  a-x 

x  +  a-1 

20. 

2  +  a  +  6 

2-a  +  6 

oi 

3a;  +  4a. 

Xll 

AUSWE 

UiS. 

22.^-2. 

24.  x^  -  y\ 

26.  2^'  +  3^^ 
3a;2  +  2^2 

23  a: +  3. 

a;  +  2 

25.^^1, 
x-\ 

27.     /-5 

a;2  +  x  -  3 

28. 

a(^x- 

-_3). 

?,9 

Sx' 

-2x  +  3 

Z{x- 

-2) 

2x^ 

+  3a;  -  2 

3f 

J     1^  -  Ax"  \1x- 

3 

4x=*  +  3x"^  -  18a;  +  27 
Exercise  42. 


1. 

a;-2  +  -- 

X 

6.^a;2 

,,_.^... 

2. 

2x^  +  3- 

5 
2a; 

' 

7.  a;2- 

-l  +  l-«. 
a;  -  1 

3. 

2a'x'  +  1 

7  +  a 
5aa; 

8.  a;2  - 

-a;  +  2      -^(-= 
a;2  +  a; 

1)  . 
-1 

4. 

x-^-4x  +  5- 

-      ^     '                   16.  1 
a:+  1 

X  +  a;2  -  -^-  . 
1  +  a; 

6. 

x  +  2y- 

V  +  1  . 
X  +  2/ 

17.  1  - 

X  1  ^x^       ^^'  ' 

2a;* 

^      ^"^       1  +  a; 

-a;» 

Exercise  43. 

1. 

a^  -  a  +  1, 
a 

6.      ^-^     . 

a;2  +  a;  +  1 

11.  ^i±^. 

a;  +  a 

2. 

x-" 
x-1 

^   a2  _  a.2  +  ^  +  1 

a  +  a; 

•       12.  -^  • 
1  +  a; 

3. 

x^  -2x 
X-  1 

o   1  -  2a  +  a^ 
^'          2a         ' 

13.   ^-1. 
x  +  1 

4. 

2a; +  1 

a-1 

14.  ^'-*. 

x'  +  l 

6. 

a2  +  ab 
a +  26 

10.  «'  +  i- 

a  -  2 

16.   1  +  2^-^. 
1  +  a; 

Exercise  44. 

15.     ^    • 
a;  -  1 

1. 

4a;     15a;  , 

18'    18 

2    *^4« 
^'   106 

76      10a 
'    106'   106 

ANSWERS.  xui 


a         4b        lab  g        1         3a; -3. 

1  2a2  -  2a       3a 


9. 


a?  —  a      o?  —  a  ^  o?  —  a 


11  a;^  +  a;''  +  a;  x  +  l 

'  {x+  l)(ar''-l)'  (a;  +  1)  (a;3  _  1) 

12  a;  a;(2a;  -  3)        4a;''-  9 

*  a;(4a;''  -  9)'  a;(4a;2  -  9)  '    a;(4a;2  -  9  * 
2a;  +  4      15a;  -  30  18 


18. 


6(a;2-4)'  6(a;2  -  4)  '    6(a;2  -  4) 


21. 


{x  + 1)^  12(a;  +  l)(a;-2)(a;  +  3)         (a;-l)(a;-2)       . 

(a;  +  l)(a;-2)(a;+3)'  (a;  +  1)  (a;-2)  (a;  +  3)'  (a;  +  l)  (a;- 2)  (a;+3) 
22   2(a  -  6)^     abja^  -  h'^Y      (a  +  by      4(a^  -  6')    _  ^6_     . 
(a^  -  bY  "  (a'  -  6')'    '  (a'  -  &')''    («'  "  6')' '  («'  "  6')' 


l.ii 


6a; 

8a;  -  9  +  12a 


12aa; 

156 

-4c- 

6a 

6a6c 

4a;- 

-7 

4a; 

2a^ 

-\-  ab  - 

-362 

6ab 

3a=» 

+  b 

6a^b 
jg  3w2  -f  1 


20. 
21. 


(m  +  1)  (m  -  1)2 
x^  +  2a;  -  1 


Exercise  45. 

7    9  -f-  10aa;2 
'   ,  12aa;2 

,o   25a -206. 
*^-         12 

Q   2a;  -f-  3 
^'      30     ' 

14.  8x  +  65  . 
21 

9.3a;+l. 
24 

6xYz 

-^^• 

Ifi    «'  +  »'. 

"■r^- 

17.            1.-..    . 
7a;  -  x"  -  12 

12.   7-«^. 

-A- 

24. 

1  +  a 

. 

a;'-  1 

a;" 


25. 


9-a» 

1 

8a;»-2 

4a; -1 

'^  -  4  26. 

a;»-l 


22    2^'  +  3a;-l. 

a;(ar»-l)  ^7  a;'  -K  5a;  -HO 

23.  0.  *  (a;  +  1)  (a;  4-  2)  (a;  -h  3) 


xiv  ANSWERS. 


M^jL?) 30. 


{2x  +  1)  (2a;  -  1)  {x  +  1)  (2a;  +  1)  (2a;  +  3)  (a;  -  1) 

29   _^L_.  31   5a;'.y  -  3.y^ . 

'  (a  +  6)3  *  a;(a;2  -  y^) 

32.  0.  3^   44  -  9a; .  3^3:2  +  qq^.  _  9 


33  a:'  +  4a:  -  13  .  ^  +  ^^  ^(a:^  -  9)  (a:  -  3) 

2(a:'^-l)  36  3a:  +  2  38—1—- 

34.  0.  •  {x'  -  1)  (a:  -  2)  *  a:(a:»  +  1) 

39.  — ^^— •  40.  ^  +  1 


a:^  —  1  x^  —  X 

Exercise  46. 
1.^^.  4.       *  7.0. 


1-x^ 

a-Sb 

a'-b^ 

Zxy 

a^-b^ 


8.        1^ 


2-^i-^-  5-0.  8(1 -a^) 


3.  _^^_.  6.  ^~^^'-'  9. 


4^2  _  3.2 


1  -  4a'^  1  -  a:3 


10. ^^l-^: ..  14.         -7 


(a:  -  2)  (a:  -  3)  (a:  -  5)  12a:(a:  +  1) 

11.  5-46 j5 :c 

(a  -  3)  (a  -  2)  (6  -  2)  3(a:2  -  9) 

j2    7ft'  +  19a  .  16    ^'  ~  15^  ~  1^ 


12(ft2  -  9)  {tP-  -  9)  (a;  -  1) 

13.  0.  17.  0. 

jg^  17x^j^i2x+_39.         20.0.  21.1.  22.0.         23.1. 


15(a:»  -  9) 


Exercise  47. 


10. 


a-1 

a{x  +  1) 
2a: +  3 

1    26»^.  g     3(a:  +  1)  . 
*  Zacy  '  x{2x  —  1) 

2.  9^.  7    ?a?.  11 
4x^  '    X  '  S{3x  -  1) 

3.  i.  g      ab      .  j2   (2a:  +  1)'  , 

4.  1.                             '  2a  -  1  '    (a;  +  1)» 

5^      5.v^g  .  g    a.-'  +  2a:  -  3 .  ^3      a  4-  a; 

7                       'a:  '  x\a-x)' 


14. 

a'  +  a  +  1 

a 

15. 

(3a: -2)'. 
(2a:  -  3)2 

16. 

x'-\ 
x^-1 

17. 

2 
x+l 

18. 

x 

a+  1 

2a-  1. 

a-  1 

4a:'  +  2a;  +  1 

ANSWERS,  XV 

20.  -^ •  27.  ^  "*"  ^~  ^ . 
3a6  a  —  x  +  1 

21.  1.  28.  1. 


23.  1. 

24.  1. 


x^ 

-x-1 

a:' 

-x  +  S 

x^ 

-Ax +  9 

2(1  -  5a:) 

3         2(1  -  5a:)  g   3(4  -  3a;) 

*  (2«+3)(2a;  +  l)*        *  4(2  -  a;) 


29.  X. 

a  +  b  +  o 


22  • 

'  X  +  y  30  «'c  +  «6'  +  60* 


31.  1. 

32.  i 
26  a 


x^-xy  +  y^                  25.       ^  ^^  ^^  +  4^^ 

19-  1-                                    26.  x  +  y.  '      4mn 

Exercise  48. 

1.  2(2  -  a:)  .                 ^^  «-l.  jg    a;-a  +  l. 

a;                             '  a  +  I  '   x  +  a  —  1 

2       ^      -                     n    2a -1  20.  -i 

2^+1                                 a      '  21.  0. 

3.  a:  +  1.                        12.  (a  +  1)2.  22.  a  -  1. 

4.  ^                            '             ^  +  1  23.   ^^  -  e^  +  1 

^^'-^(^TZ)  ab-cd-l 

^  ~  ■  ^                         14.  a  +  a:.  '^^- ~- 


26c 


15.  «(^--.tJl  .  o.    1 


6.-^.  ^^--.Tl^-  25. 


X 


-    4a;'  +  2a;  +  1 .  jg   _J__.  26.  2a;. 

2a;  '   2a:2-l 


27.  (^  -  y) 

8.1,                            17.  ^±i^.  ^ 

a:'                                    a;  -  y  28.  - 1. 

0  §. .                              18    <^<^  +  ^^  ~  <^^  .  29        1      , 
'  X                                 '  ac  +  bo  +  ab  *  1  +  2a; 

Exercise  49. 

1  (^  ^  1)'    .                4        8na;      .  y       14a; 
'   in"  -  x^'  '  9a:2  -  4 

2  ^  -^^o  .              5   ^{2x_.z_S),  6 


2(3a:  -  1)  2-z 

9.  a 


:vi 

^JVSfTT^iJ^'. 

' 

10.  *- 

13.          ^'         . 

2a(6^  -  a») 

"•x(2. 

9 

-a:)(; 

J.            \%x  +  3 
x-Z)                        •(9x^-l)(4-' 

9a;') 

''■a'\i 

^^•(T 

4 

-x){x-1) 

(0^-3) 

17.  1. 

13      1  +  2^ 

21. 

22.  • 

4(1 -a').          25. 

26. 
_^. 
^■'                    27. 

hx. 

X 

x^^y\ 

30.  ^r^-> 
2a; -r 

•  X{1  +  x^) 
20.  1. 

1 

23. 
24.  : 

-f- 

a +  6.                29. 
a-6 

a  +  h 
a-b 

X 

x  +  1 

32.  «. 

a; 

33.  ia. 

34.  6  -  3» 

35.  X. 

37.  -  f . 

QQ     M 

2  +  52^2 

^^'     d^ 

+  Zb^c^ 

OR  5a- 

1. 
1* 

38.  -  X. 
Exercise  50. 

40.^. 
a 

- 

1.  6. 

13. 

V- 

25.  -  2. 

37.  tV 

49.  2. 

2.  2. 

14. 

|. 

26.  f. 

38.  5. 

50.  f . 

3.  3. 

15. 

-i 

27.  0. 

39.  2825. 

51.  -  i 

4.  -2. 

16. 

i- 

28.  1. 

40.  .00025 

.      52.  -  ii. 

5.  10. 

17. 

-tV 

29.  2. 

41.  -.04. 

53.  0. 

6.  2. 

18. 

¥• 

30.  5. 

42.  4. 

54.^. 

7.  2. 

19. 

-¥. 

31.  -  5. 

43.  13. 

55.  -5. 

8.  -  1. 

20. 

-2. 

32.  3. 

44.  -  A. 

56.  -  7. 

9.  5. 

21. 

5. 

33.  10. 

45.  f  f 

57.  -  3. 

10.  i 

22. 

¥• 

34.  -  0.8. 

46.  H. 

58.  -If 

11.  -  9. 

23. 

-f. 

35.  H. 

47.  -  7. 

12.  -  f. 

24. 

-2. 

36.  -  5. 
Exercise  51. 

48.  4. 

1   -f 

6. 

-\' 

11.  -  23. 

16.  8. 

21.  0. 

2.  -3. 

7. 

-i 

12.  -  0.5. 

17.  0.4. 

22.  -  3. 

3.  -if 

8. 

-2. 

13.  -  0.1. 

18.  I 

23.  f 

4.12. 

9. 

-f. 

14.  -  0.1. 

19.  5. 

ft.  -  9.  10,  -  f  15.  if  20.  7. 


ANSWERS.  XVii 

Exercise  52. 

1.  3a.  g  a-  b  «  a.  jo    a* - 6* . 

2c  *  6  '    g»  +  6' 

2- 5*  6.1^.  10.^^2.         14.0. 

^-2«  ^«7'  15.17a. 

4.  -A^.  8.    -«^.  12.  «.  17.  ^. 

a-b  a-b  2  3 

18.  «6cd  22.  gc»  27.  ^1  -  2a  -  a*), 
ab  +  6c  +  ac              '  gc  -  g6  +  be  28.  -  a. 

19.  1.  2^-  ^-  29       ^^      . 

2a -36 


22. 

ac» 

ac  -  ab  +  be 

23. 

0. 

24. 

g3  +  g 
3a2-l 

25. 

c^. 

Oft 

ab 

20.  ^-^;-  ^«'  -  A  30         36g6 

2a  +  6  orr    ^2  •   2262  _  15^2 

2j        a6(a  —  6)  ofi         g6  o^   a^  -  a"  -  a  +  1 

•  a»-2a62-6'  *         *  2{b^  -  a')'  '  a 

Exercise  53, 

1.  24.  4.  63.  7.  33  and  42. 

2.  45.  5.  27 ;  28.  8.  $12,000. 

3.  60.  6.  48  ;  49  ;  50.  9.  144  trees. 

10.  26  ;  27  ;  28.  15.  13  years. 

11.  A,  $32 ;  B,  $48 ;  C,  $50.  jg     f  A,  21  years. 

12.  16  and  81.  *    *-B,  35  years. 

13.  21  and  79.  jy     f  Father,  48  years. 

14.  14  and  54.  '   \  Son,  20  years. 

18.  A,  $960 ;  B,  $1200;  C,  $1080 ;  D,  $1760. 

19.  70  acres.         23.  \\  days.         26.  2f  days. 

20.  44  and  45.        24.  6  days.  27.  4  days. 
22.  3f  days.         25.  12  days.         28.  36  min. 

29.  169tV  min.  33     f  32^^  min.  past  6. 

30.  20  days.  *    ^  54y«x  min.  past  10. 

32    /  ^^ri  min-  P^*  ^-  34    /  ^tt  and  38,2^  min.  past  4. 

t  38^T  min.  past  1.  '    *-  21  j»y  and  54^^  min.  past  7. 

36.  24  hours.  38.  6  hours.  40.  12  mi.  an  hr. 

37.  208  miles.  39.  55  miles.  41.  19^  miles. 


xviii  ANSWERS. 


42. 


Ist,  each  VZ  hrs.  54.  108  and  72. 

A,  42  mi.;  B,  40  mi.  55.  30  apples. 

2d,  each  492  hrs.  56.  334  pages. 

A,  1722  mi. ;  B,  1640  mi.  57.  4  and  16. 

44.  Hound,  150 ;  hare,  250  leaps.  58.  81  yards. 

45.  Hound,  72  ;  hare,  108  leaps.  59.  A,  4  days ;  B,  5  days ;  C,  6  days. 

46.  $63.  60.  5t\  and  38y\  min.  past  10. 

47.  1713  men.  61.  Dog,  600;  fox,  900  leaps. 

48.  45  men.  62.  19^^  days. 

49.  2160  men. 

go     r46|bu.  oats. 

1  53i  bu.  com. 
gj     f  $3250  at  4%. 

I  $1800  at  5%. 

52.  10  ;  14  ;  6  ;  24. 

53.  First,  20  days;  sec,  15  daj^s.  "    b  -a 

Exercise  54, 

1.  $156.  4.  $264.  7.  8%.  10.  $540. 

2.  5^  yrs.  5.  10  yrs.  8.  $333}.  11.  8  months. 

3.  4^%.  6.  4|%.  9.  25  yrs.;  16  yrs.  8  mos. 

Exercise  55. 

1.  a;  =  1.  5.  a;  =  2.  9.  x  =  |.  13.  x  =  15. 

y  =  i.  y^-^'  y  =  \'  y  =  10. 

2.  a;  =  1.  6.  a;  =  1.  10.  a;  =  -  \.  14.  x  =  3. 
y  =  -\.              y  =  -h                 2/  =  2.  ^  =  -4. 

Z.  x  =  l  7.  x  =  -S.  11.  a;  =  3.  15.  x  -  10. 


63. 

b  +  c 

miles. 

6<1 

ab  +  ( 

G    a  - 

c 

6  +  1 

'  6  +  1 

65. 

abc 

feet. 

y  =  -h  y  =  H'  y  =  -7.  y  =  -io. 

x  =  2.  8.  a;  =  |.  12.  x  -  8.  16.  x  =  12. 

2/  =  3.  y  =  -h  y  =  9'  y  =  is. 


Exercise  56. 

1.  a;  =  1.  5.  a;  =  -  2.  9.  a;  =  -  3.  13.  a;  =  f 

y  =  l.  y  =  h  .V  =  -  4.  ^  =  -  4. 

2.  a;  =  -  1.  6.  a;  =  3.  10.  a;  =  12.  14.  x  =  l 


y  =  -i. 

y  =  -h 

y  =  12. 

y  =  -h 

3.  a;  =  2. 

7.  x  =  0. 

11.  a;  =  6. 

15.  a;  =  4. 

y=--i. 

y=2. 

2/ =  20. 

y  =  -s. 

4.  a;  =  -  3. 

8.  a;  =  -  2. 

12.  X  =  15. 

16.  a;  =  -  21. 

y-0. 

y  =  -3. 

y  =  10. 

y  =  -  40. 

10. 

X  = 

=  7i 

y  = 

=  - 

2J. 

11. 

X  = 

=  i 

y- 

=  i. 

12. 

X  = 

=  4. 

y  = 

=  5. 

ANSWERS.  XIX 

Exercise  57. 

1.  a;  =  1.  4.  rr  =  3.  7.  x  =  6. 

y  =  1.  2/  =  -  1.  y  =  6. 

2.  X  =  -  1.         5.  a;  =  |.  8.  a;  =  12. 
y  =  1.                  y  =  f.  2/  =  12. 

3.  x  -  2.  6.  a;  =  3.  9.  x  =  12. 
y  =  -  2.              2/  =  -  2.  y  =  35. 

13.  a;  =  0. 
2/  =  3. 

Exercise  58. 

1.  X  =  5.  5.  a;  =  3.  9.  a;  =  2.  14.  x  =  18. 
y  =  12.                2/  =  1.                     y  =  4.  ^  =  12. 

2.  a;  =  5.  6.  a;  =  1.  10.  a;  =  -  0.2.  15.  a:  =  9. 

y  =  2.  2/  =  1.  y  =  0.6.  .y  =  -  1. 

3.  X  =  i  7.  a;  =  3.  11.  x  =  .015.  16.  x  =  17. 
2/  =  i.                 3/ =  5.                    ^  =  .01.  y  =  6. 

4.  a:  =  7.  8.  a;  =  1.  13.  a;  =  2.  17.  a;  =  2. 

y  =  5.  y  =  -l.  ^  =  -3.  y  ==  -  l^ 

18.  a;  =  -  2. 
y=-3. 

Exercise  59. 

1.  a;  =  2a.  6.  x  =  a  +  26.  10.  a;  =  w  -  m. 
y  =  -  a.                       y  =  2a-b.  y  =  n -\^  m. 

2.  X  =  —  b.  „  en  r-  bd 


11.  a;  =  3. 


=  2a.  *  an-  bm  ,„  _  2a  +  1 


b'  -  b  ,,  _  «d-_cm , 


^  6 


^- ^      ab'-a'b  an-bm  yi,  x  ^ 


a-vb  -^  G 


y-    «-« 
-^       ab'  -  a 

'6* 

«-  =  !• 

V               ^         ■ 

""      a  +  6  +  c 

4.  X  =  m  +  n. 

.,  =  *. 

13. 

X  =  a  ^  b. 

y  =  m  —  n. 

a 

y  =  a-b. 

5.x      26  +  1. 
b 

9..  =  1- 
c 

14. 

a-  d 
b-d 

-"-^^- 

-i- 

-H- 

15.  a;  =  a. 

16. 

a; 

=  -  a. 

17. 

X  = 

b. 

18.  a;  =  a  +  1. 

y-6. 

y 

-6. 

y  = 

a. 

y  =  6  -  1. 

XX 


.  \ 

Exercise  60. 

1.  x  =  1. 

3.  a:  =  3. 

5.  X  =  1^. 

7.  a;  =  -  3. 

y-2. 

2/  =  4. 

y  =  li 

y  =  3i 

z  =  3. 

0  =  7. 

z-=lh 

z=  -2. 

2.  a:  =  2. 

4.  a;  =  3. 

6.  a;  =  2. 

8.  u  =  2. 

y  =  3. 

2/ =  -2. 

y  =  3|. 

v  =  3. 

z-  -  4. 

2=  -4. 

2  =  -4. 

w;  =  1. 
x  =  4. 

9.  x=-  12. 

10. 

a;  = 

a  +  6. 

11.  x  =  6. 

2/ =  18. 

y=- 

a-b. 

y  =  40. 

0  =  -  24.  2  =  2a.  2  =  20. 

12.  a;  =  -  a  +  6  +  c.  14  a;  =  ^^  ~  2& . 

y  =  a  ~  b  +  c.  '  6 


2;  =  a  +  6  —  c. 


2a  +  36 


13.  a;  =  a  -  6  +  1.  6 

y  =  —  a  +  6  +  1.  g^  g  +  6. 

0  =  a  +  6-l.  6 

Exercise  61. 

1.  a;  =  ^.  4.  a;  =  —  ^.  1.  x  =  \.  9.  a;  =*  a. 

y  =  -\.  y  =  i  2/  =  -i  y=~-ck 

2.x=-l,  5.a.  =  f.  8.:r  =  -2^.  10..  =  1- 


3/  =  1  2/  =  f  •  *  1  +  n' 


m 


3.  a:  =  f  6.  a;  =  i.  ^  _     2n     .  1. 

11.  a:  =  ?.  17.  a:       "  ^60  . 


6  6  +  c 

2ac 


a 

12.  a:  =  1.  2  = 
y  =  l. 

13.  ^  =  |.  13   ^  _      2a 

14.  a:  =  l:  2/  =  -i;   0  =  i  .V  =  r^ 
\b.x=2;y=-\',z==l. 


16.  x  =  i;y  =  i;2;  =  i.  w+n 

19.  a;  =  i;  y  =  -|;  ;s  =  l. 


ANSWERS,  XXI 

Exercise  62. 

1.  9  and  14.  2.  9  and  12.  3.  2  and  8.  4.  f 

6.  Flour,  3  cts. ;  sugar,  5  cts.  13.  A,  91  years ;  B,  30  years. 

6.  f .  14.  84  and  60. 

7.  49.  15.  A,  4|  miles ;  B,  4  miles. 

8.  Man,  $3 ;  boy,  $2.  16.  12  boys,  $60. 

9.  y*j.  17.  Length,  8  in. ;  breadth,  6  in. 

10.  57  pe?r  trees ;  43  apple  trees.        18.  A,  in  24  days ;  B,  in  48  days. 

11.  Sheep,  $4  ;  calf,  $7.  19.  A,  $70 ;  B,  $110. 

12.  A,  $660 ;  B,  $480.  20.  480  miles. 

21.  j\.  24.  f  27.  24. 

22.  11  and  36.  25.  23.  28.  64. 

23.  i^  and  |.  26.  56  and  65.  29.  253. 

30.  151.  37.  i  and  f. 

31.  Silk,  $1.80 ;  satin,  $1.50.  38.  A,  $26  ;  B,  $14 ;  C,  $8. 

32.  16  ;  20  ;  24.  39.  A,  $70 ;  B,  $50  ;  C,  $90. 

33.  15  gals,  from  1st;  6  gals,  from  2nd.  40.  8  men  ;  6  women  ;  10  children. 

34.  $600  ;  eldest,  $200.  41.  6  doz.  at  30  cts.;  3  doz.  at  40  cts. 

35.  A,  $40 ;  B,  $50 ;  C,  $80.  42.  12  yards  by  8  yards. 

36.  8  dollars ;  40  halves ;  36  quarters.  43.  15  ft.  by  6  ft. 

44.  Fore  wheel,  5  yards  ;  hind  wheel,  6  yards. 

46.  6f  miles  an  hour.  48.  1^  mi.  an  hour. 

47.  32  miles  ;  5  mi.  an  hour.  49.  24  bu.  from  1st ;  16  bu.  from  2nd. 

50.  A,  60  yds.  a  min. ;  B,  80  yds.  a  min. 

61.  A,  27  mi. ;  3  mi.  an  hr.    B,  30  mi. ;  5  mi.  an  hr. 

62.  A,  9 ;  B,  12 ;  C,  8  hrs. 

64.  A,  4f  yds.  a  sec. ;  B,  4}  yds.  a  sec. 

55.  A,  5^  min. ;  B,  5^  min.  57.  C  helped  6  days.    A,  in  45  days. 

66.  A,  4 1  min. ;  B,  4^f  min.  58.  A,  $35  ;  B,  $26  ;  C,  $20. 

Exercise  63. 

2.  H.  C.  F.  =  2a;  -  3.  6.  ^'  "^  ^  "^  ^ 


3.  iV. 

x{x-l)*'  '  x'-l 


-x-1 


7   ^(^^  -  ^^) 
*•  A-  *   6(2a;  -  3) 


9. 

a 

'  +  x* 

a} 

10. 

X 

=  -2. 

11. 

X 

=  -f 

12. 

X 

=  -8. 

13. 

X 

=  4. 

xxil  ANSWERS. 


17.  X 


14.  a:  =  f.  lb.  x=\.  16.  a:  ==  7.  « -  & 

2^  =  |.         ■     y  =  \'  y  =  10.  y  =  —^ 


a-\^  b 


Exercise  64. 


1.  a;  <  1.  b.  X  >  6.  9.  a;  >  6  and  <  7. 

2.  a;  >  3J.  6.  a;  <  ^.  10.  a;  <  2  and  >  \\ 

3   a:>^.  7   a:  <  — «^  •  ^^^  1^,  18  or  19. 

^-  "^  >  a  ^-  "^  <  '^^r^Tb  12.  13. 

4.  a;  >  2.  8.  a;  <  6. 

Exercise  Q^, 

10.  27ar'^'.  25.  a'  +  3^  +  ^9^. 

11.  -  8a;«.  26.  a:*  +  3a;V  +  f^^ 

18.  \\xY'  27.  ^  -  1  +  ^• 

33.  fa;*  -  2x^y -^  f fa: V  -  \xy^  +  j\y*. . 

Exercise  66. 

1.  a»  -  3a26  +  Sab^  -  bK  3.  1  -  4a;  +  6a:2  -  43:3  +  a:*. 

2.  x^  +  Sx^  +  3a;  +  1.  4.  a^  -  6a^  +  12a  -  8. 

6.  16  +  32x2  +  2ix^  +  8a;«  +  a^, 

6.  a*  -  10a*b  +  40a362  _  30^253  ^  gOaft*  -  326*. 

7.  a;''  +  15a;*  +  90a;3  +  270a;2  +  405a;  +  243. 

8.  a«  -  6a*6  +  UaW  -  86». 

9.  32c5  -  80c*c?2  ^  30^3^^*  _  40c2(i«  +  lOcrfs  _  (^lo. 

10.  a*  -  12a'62  +  54a26*  -  lOSaft^  +  8168. 

11.  343  -  441a;2  +  189a;*  -  27a;«. 

12.  a;*2/8  +  8a;''^«  +  24a;2^*  +  Z^xy"^  +  16. 


32 


13. 

8  + 

12a;  ^  6a;' 
a          a^ 

-+^3- 

a^ 

14. 

243 

405c*^  ^ 
2 

136c* 
2 

45c6 

4 

^  15c8 
16 

15. 

1  - 

2c2      3c* 
b        262 

263 

C8 

166* 

16. 

32a;« 

+  5c*    ^ 
'      16a;* 

5c«    ^ 
4a;3 

5c2    ^ 
2a;2 

2a; 

ANSWERS. 


ZXlll 


17.  a*  +  Zx^-b3^  +  Sx-l. 

18.  2^-9x^  -^  24a:*  -  9x^  -  24x^  -  9x  -  1. 

19.  a8  +  4a^c  +  iOa^c^  +  IQa^o^  +  19a*c*  +  IGa'c^  +  lOa^c*  +  4ac'  +  c«. 

20.  a:»  -  3a:V  +  Sx'^z  +  '6xif  -  &xyz  +  ^xz"  -  y^  -\-  ^y'^z  -  Zyz^  +  7^, 

21.  8a:«  -  12a:*  +  42x*  -  37 ar»  +  GSx^  -  27a:  +  27. 

22.  1  +  4a;  +  2a:^  -  Sar*  -  5a:*  +  Sa:^  +  2a;6  -  4a:^  +  x^. 


1.  Zxy\ 

2.  5a». 

3.  122/«. 


Exercise  67. 

8.  3a:^ 

9.  hy^. 


10. 


\ab''. 


15.  -  8a:». 

16.  2^». 

17.  -  Ix^y, 


Exercise  68. 


1.  a;2  -  2a:  +  1.      2.  1 


3.  3a:2  -  2a:  +  1.      4.  5  +  3a;  +  x*. 


5.  ri'  -  2n'  +  3. 

6.  2x^  +  3a:'  -  2a;  -  3. 
13.  ^a;  -  5. 

16.  a:'  +  X  -  ^. 

17.  ^d'  -  ia  +  6. 

18.  ^  +  3+^- 


19.  |a;2  -  fa:  +  f. 

24.  1  +  2a:  -  2a:2 

25.  1  -  a  -  ia'  . 


26.  a:  -  ^  -  A 


_9^ 
2a:3 


27.a+26_2^' 


Exercise  69. 


1.  85. 

4.  325. 

5.  427. 

10.  90.08. 

11.  14.114. 
14.  0.17071. 
16.  2.6457  . 


17.  3.3166 

18.  3.5355 

19.  1.8257 

20.  1.4529 

21.  0.9486 

22.  2.5819 

23.  1.2747 


24.  0.3415 

25.  0.2213 

26.  1.0031 
6.0075 
1.9318 
1.1117 
1.3687 


27. 
28. 
29. 
30. 


Exercise  70. 


1.  a  +  2a;. 

2.  3  -  a. 

3.  1  -  4a;. 

4.  a'  -  a  -  2. 

5.  a:'  -  a:  +  1. 

6.  1  -  3a:  -  2a:'. 

7.  4a:'  -  3a;  -  2. 


8.  a'  +  5a  -  1. 

9.  2a:'  -  5a;  -  3. 
10.  2  -  3n  +  3n'. 


11.^ 


h 


12   ^  -  -%. 
2y       3x 


13.  a;  -  1  +  -  - 

X 

14.  1  +  -  -  -^ 

a      or 


15.  a;'  +  — 


2^^ 


XXIV 


AmWEHS, 


Exercise  71. 

1.  16. 

3.  124. 

5.  3204. 

7.  70.09. 

2.  91. 

4.  352. 

6.  804.6. 

8.  0.0503. 

9.  0.997. 

13.  1.542  . 

• 

17. 

0.2147  .  .  .  . 

10.  4.217 

14.  1.953  . 

.... 

18. 

1.021 

11.  1.817 

15.  2.704  . 

.... 

19. 

2.0033  .  .  .  . 

12.  1.775 

16.  0.3968 

20. 

2.901 

21.  1.730 

22.  0.0535 

Exercise  72. 

1.  19. 

7.  2?- 

_^. 

11. 

1.5704  .  .  .  . 

2.  43. 

y 

2x 

12. 

2; -3. 

3.  3.08006  .... 

4.  0.9457 

8.  2  +  2a:  -  x\ 

13. 

2a- i. 

5.  1  -  '6ab. 

9.  14. 

a 

^.x-l. 

10.  34. 

14. 

4a:' -i. 

Exercise  73. 

4.  21^0: 
6.  2a}^y¥. 

30.  81. 

31.  64. 

38.  a^x\ 

44.  a^. 

7.  Va^m^. 

32.  36. 

39.  3a%. 

45.  1^2ir. 

24.  9. 

33.  -  27. 

40.  4x\ 

46.  a^y. 

25.  125. 

34.  if. 

ij- 

47.  2x\ 

26.  8. 

35.  -W-. 

41.  a^. 

27.  16. 

28.  128. 

^*    32 

36.  ak 

37.  2ai 

42.  7ai 

48.   2«^^\ 
c 

29.  4. 

43.  2a:i 

49.  ,Vir. 

Exercise  74. 

7.  3  X  2-^ac-\ 

1 

OK    750 « 

34.  f. 

43.  i 

13.^. 

25.            ♦ 
ca:V 

35.  i 

44.  2a. 

x^ 

36.  216 

50.  3ai 

51.  c'dl 

14.-^. 
0 

27.  i 

28.  i. 

37.  f  ^. 

38.  -  f 

20.  ^^f. 

a?e 

29.  32. 

30.  25. 

31.  h 

39.  ^h. 

40.  -  -i 

I- 

52.  m\ 

24IJ5M. 

32.  ^i^. 

41.  tW 

63.^- 
x^ 

a^x 

33.  54. 

42.  1. 

ANSWERS.  XXY 


62.  «^.  7aio  •     i 

I  66.  -^  •  68.  X  ="*  . 

6a  ^.  j,^ 


Exercise  75. 

12.  A.  26.^.  31.2^  40.0^. 


13. 


o*  c^*  ^f  41.  27. 

x^.  27.-^^-  32.1.  42.4 

25  i9!i^l  33.  Va.  ^ 


125a; 


34.  Va.  jQ   ^i 


,,      1  2  ^4.  Va.  43.  ai5c« 

•  4a^  28.^.  o.   A.  ..     a» 


15. 


J 


44. 


^.  9x^  ^y  *  1/5 


24.     1 


^  ^1^  36.       ^' 


«¥  y^  '  37.  ,j^. 


«^^»V         4^-  it. 


c^& 


25.^.  30.  -i^.  38.  (f)l  ^^-"fed' 

2/^  «*  39.  -l^i  47.  xyz, 

48.  8a:3  -  36  +  54a:  -  3  -  27a;  -  ». 

49.  a:2  -  8a;'^^^'  +  24a;^  -  32a;^  +  16a;t 

60.  243a;^  +  810a;''^'  +  1080a;6  +  720a;"^'  +  240a:'''^  +  32a; ^. 

61.  T^a;  -  2  -  a;  ~ '^  +  6a; "  ^  -  16a;  ^  +  16a;. 

62.  16a;  -  2  +  160a;  ~  '^  +  600a; "  ^  +  1000a; "  ^  +  625a; "  i 

63.  a;2  -  4a;  ^2/  ~  ^  +  6a;y  -  *  -  4a;^y  ~  ^  +  y  -  2. 

64.  1    +    4a;~%"^   +  -^a;-'^"^  +  -¥/^~^^~'  +  f^a;-V~^  + 
fta;"^^"^  +  A^a;-^-*. 

66.4.  56.9.  57.^.  68.  f  59.  -  32r 

60.  1.  61.  -^  •  62.       ^ 


V^  •  (~  3)' 


XXvi  ANSWERS. 

Exercise  76. 

1.  2a^  -  a  +  9.  3.  9x  +  bx^y^  +  Qy, 

2.  a  +  1.  4.  4x^-1  +  12a;"  i 

8.  ex^  -  *lx^  -  19a;^  +  52;  +  9x^  -  2x^, 

9.  2-  4a"M  +  2a~%^ 

10.  5  -  3a;~y  +  12a;"^^^  +  4x^y~^. 
11.  bx^  -  3a;^  +  1.  12.  4a;  -  ^  -  3^  - 1  -  2a:3^-« 

13.  x"^- 2a:~^ +3a;"^- 1. 

18.  x^y  -  1  -  ^x^y  ~  ^  +  2  -  4x~ ^y^. 

19.  3a~^- 2a"M  + 4a~^a;-a;i 

20.  2a^  -  3a" tV  _  a  ~ TJ  •  23.  a ~  ^  -  26^  +  Sah. 

21.  x^  -  2x^y^.  28.  3a;^  -  ixy~^  -  2x~^y'K 

29.  ix'^y-  ^x~^  +  Sy-K 


Exercise  77. 

1.  21/3. 

4.  -  2V5^.               29.  }l/6. 

32.  |1/3D. 

2.  31/2. 

5.  41/6.                   30.  1/TO. 

33.  t'^t/6. 

3.  31/5. 

6.  -  61/T.               31.  ^Vm. 
Exercise  79. 

34.  :r-^l4a. 
4a; 

1.  l/I^. 

2.  1/35.         3.  1^432.         4.  -1/20. 

18.  -Vx-i 

Exercise  80. 

1.  Va.                 2.  l/a.                 5.  V^.  10.  f  3^^^ 

Exercise  81. 

1.  1^345";   l^m:                  4.  l^i;   l^f.  14.  l^B. 

2.  l^m-;   i>27.                    7.  1^64;   i^Sl ;   1^125:  18.  1^?^. 

3.  1^27 ;  1^25.                   13.  V^.  21.  1/3. 


ANSWERS. 


XXVll 


1.  5T/2; 

2.  v5. 

6.  21^. 

7.  31^5. 

11.  il/6. 

12.  51/g. 


13.  2Vax. 

14.  fl^20. 

15.  yi^i^e. 

16.  4VB. 

17.  2l^2c. 

18.  0. 


Exercise  82. 

19.  dacVE. 

20.  -  66T^2a. 

21.  8K2. 

22.  -  191/3. 

23.  -  21/6. 

24.  71/io. 


61/5; 


25.  0. 

26.  61/J 

27.  51/T. 

28.  -  51/3. 

29.  121/6  +  101/5". 

30.  a6l/3a. 


Exercise  83. 


1.  12. 

2.  15V/3. 

11.  al^aU^. 

12.  1^72. 

13.  a:1^864a^. 

14.  31^24. 


3.  121/3. 

4.  21^3. 

15. 
16. 
17. 

18. 


5.  361/6. 

6.  301/21. 

1^3356. 

61^.  y    1y 

1^1^.  20.  ^1^288: 


26.  21/&  -  41/3  +  8V5. 
29.  201/2  +  30  -  81/15. 

31.  2  -  41/2. 

32.  2  +  71/3. 

33.  -  6  -  21/6^. 

34.  16VIS  -  30. 

35.  71/6  -  12. 

36.  42  ^6V'I0  +  21/5'. 

37.  21/15  -  6. 

38.  -  282  -  721/10. 


19.  V 


7.  71/5. 

8.  tV 
6|3a5ai2~ 


9.  fl^3. 
10.  ^1^5. 

22.  i^f. 

23.  !^^ 


21 


24. 
25. 


^. 


39.  301/6  +  541/5^  -  34. 

40.  2. 

41.  96  -  161/5. 

42.  a;l/6  +  l/3a;'^  -  3a;. 

43.  l/3a:-'  +  3a;  +  a;  +  1. 

44.  21/a;'*  -  1  -  ^x  -  ^, 

46.  25  -  7a;. 

47.  2a;. 

48.  251/3". 

49.  36a:2  -  50a;  -  100. 


1.  3. 

2.  21/f. 

3.  31/3. 
10.  il/6. 


Exercise  84. 


14.  ^1^72. 
16.  3. 

16.  l^f. 

17.  1^^*. 


21.  51/7-14. 

22.  1/2  +  3. 

23.  1/21  -  51/B. 

24.  21/7  +  41/6  -  6VH 


1.  iv^. 

2.  il/6. 

3.  2Vi5 

15 


21/5 


6 


Exercise  85. 

21/7  +  1/35 


6. 


6. 


14 
2l/g  -  V\ 

4 
1^20  -  1^? 

6 


l^T2-vlB 


9. 


11 


5 

6Vl 


10.  13  +  71/5. 


xxviii  ANSWERS. 

'        13T/g-30  12    1A±J^.               13  Yl±}^. 

6  '23                          *          5 

14  6a  +  V^g?)  -  126 .  13   5  -  T/30  -  51/6  +  6t/5'. 


4a  -  96  5 

x-S 


15   rg  +  T/^Tl  -  5.  19   21/3  -  1/^21. 


1^   6a- 6  +  5l/2a^- 

a. 

20.  Vx^ 

-  ] 

[-a;». 

•*^"-        '     "           ,^                n 

14a-  9 
17.  VS  +  V2. 

21.  H«? 

+ 

W-Vab 
b 

22.  2.12132. 

25.  0.057735. 

28.  0.10104. 

23.  1.05409. 

26.  0.709929. 

29.  2.63224. 

24.  4.53556. 

27.  9.00996. 

30.  0.62034. 

Exercise  86. 

1.  w'. 

4. 

6. 

7.  8x^\ 

10.  4al/25:. 

Exercise  87. 

1.  21/2  -  3. 

8. 

31/5  -  41/2. 

15.  a  -f  31/a*^  + 

2.  1/3  +  21/5. 

9. 

21/15  -  31/3. 

16.  1/3  -  1. 

3.  3V3  -  21/2. 

10. 

^1/2  +  |V/6. 

17.  1/3  +  1/2. 

4.  VE-  1/3. 

11. 

|l/3  -  |l/6. 

18.  2  -  1/3. 

5.  1/14  +  21/7. 

12. 

f +  1/3'. 

19.  3-1/2. 

6.  31/5-21/7. 

13. 
14. 

2  -  il/3. 

20.  2  -  il/2. 

7.  2V'5  +  1/6. 

1/m  +  n  +  l/m 

-n 

'.    21.  1  +  \V^. 

22.  1  +  V% 

Exercise  88. 

1.  8.             5. 

2. 

9.  -|. 

13. 

1.               17.  i 

2.  4.             6. 

-i 

10.  -  1. 

14. 

9.                18.  \. 

3.  3.             7. 

8. 

11.  12. 

15. 

-V-.             19.  %. 

4.  2.             8. 

14. 

12.  -2/. 

16. 

64.             20.  ^. 

21.  18. 

28. 
29. 
30. 
31. 

1. 
8. 
1. 

35    ^^- 

ly 

41.  a', 
f)        42.  f. 

22.  5. 

23.  1. 

24.  9. 

25.  9. 

35.         ^ 

36. 

4a(l 

I 

+ 

26    («-*)'. 
2a -6 

32. 
33. 

9. 

37.  16a. 

38.  I 

.o       20a> 
^^'  25 -4a- 

27.  |. 

34. 

64. 

39.  i. 

44.  -  3. 

45.  ^.  46.  -  6. 


-       ANSWERS. 


Exercise  89. 


XXIX 


2.  VS. 


3.  1/5. 


4.  VS. 


5.  -^1/5E 
4a 


6.  2  +  4VS. 
1.  -  2Vax. 

8.  f V"3D  ;   fi^^IOS: 

9.  3V2  +  1/3  -  31/5  +  VlD. 
10.  1^1/15  - 1/6  +  21/m 

54  -  381/2  .         16.  2.88675. 
17.  0.18362. 
15.  4  +  Vl5.  18.  0.218286. 

X  — 


11. 


12. 


13. 


X  -Vx  -  1. 

9  +  51/?. 
6 

21/3 


1/2 


14 


7 
4  +  VTd. 

25.  31/5  -  21/2. 

26.  41/5  -  41/2. 

27.  31/7  +  21/n. 


28. 


29. 


33.  I. 


34. 


1/a; 

V-.  35.  25. 

Exercise  90. 


19.  0.36452.         22.  31^. 

20.  21/7.  23.  l^Ti. 

21.  l^'gg.  24.  5  +  21/2, 

30.  I  (a;  -  Vx'  -  36). 

31.  21/^2^^. 

32.  6. 
36.  4.  37.  f. 


9.  5l/=T. 

10.  9l/^=T. 

11.  20V^=T^. 

12.  6V^^. 

13.  81/"=^. 

14.  0. 

15.  al/^^-6. 

16.  -  (a  +  36)1/^=T. 
17.-1/2. 
18.  -  18. 
X&.  10. 

41.  -?/6  +  2V^vSl/' 

48.^  +  ^^^ 


-  7. 
-61/6^. 
91/2. 
281/6. 

-  101/ro. 

(y  -  a;)l/"=T. 

26.  a(a-J)2l/-  1 

27.  3  +  1/2. 
28.-6-5 1/6. 

29.  24. 

30.  -  19  -  21/35. 
T^.  42.  - 

l-4l/"="3 


31.  46  +  21/^. 

32.  -  2  -  21/3. 

33.  a;2  -  4a;  +  7. 

34.  6'  -  a\ 

35.  x^-2x  +  2. 

36.  x2  -  a;  +  1. 

37.  1/^=^. 

38.  1/6^. 

39.  -  31/5. 

40.  -  AaV^^, 


46. 


47. 


11 
fl'  -  6^  +  2a6l/^^ 

aM-  62 
24  +  71/10 


48.  liSV^HI. 


53:  iV^  +  31/^=^. 

54.  21/^^3  -  31/^^^. 

55.  4  -  31/-  6. 


14 
3  -  V~=^. 
VS  -  V^^. 
60.  2  -  21/"=^. 

62.  |i/^:T:+4;fl/"=T-i. 

63.  -  V-. 


XXX  ANSWERS. 


Exercise  91. 


1.  i4.  6.  if.  ,.    -.J^EI.       iA   -.26 


11.   A=J^E^.  14.   ±: 

2.  i2.                   7.  ±5.                            Af    a  a 

3.  ±^.  8.  ±1.  12.  i3a. 

4.  ±2.                    9.  ±f.                  j3    ^e.  1^-  =^(«  +  &)- 
6.  ±  |.                  10.  ±  a.                   '  ■      2  '  16.  ±  1. 

Exercise  92. 

1.  2,  -  16.           10.  h  -  |.              19.  h  -  |.  28.  1,  -  -^. 

2.  2,  -  12.           11.  I,  -  f .             20.  2,  5.  29.  ±  1. 

3.  -  2,  10.            12.  -  I,  |.             21.  3,  -  |.  30.  3,  -  |. 

4.  6,  -  1.              13.  3,  |.                  22.  3,  -  1.  31.  5,  -  ^. 

5.  -  3,  -  8.         14.  5,  -  |.             23.  3,  -  f .  32.  3,  -  f. 

6.  1,  -  |.             16.  7,  -  -1^.           24.  3,  -  I  33.  2,  -  5. 

7.  2,  -  |.             16.  f,  -  |.             25.  4,  -  |.  34.  i,  |. 

8.  -  1,  1.             17.  f,  -  f             26.  1,  -  -V-.  35.  5,  -  f. 

9.  2,  -  -V-.            18.  1,  -  f .              27.  -  2,  -  ^.  36.  -  1  ±  V^. 

gg    -3=fcl^29  .             ^j  -  5  i  i/:r-7 


43. 


o,   5  =t  1/13 

^''          6 
oo    3il/5" 

38.        ^        ■ 
,   1  ±  2V-  1 

40.  g'^'^^  42.  ^  *  '^^IT^ 


1.  2a, 

-ea. 

2.  36, 

-76. 

3.  2c, 

-5c. 

4.  ab, 

-6a6. 

n     36 

'•   2  ' 

46 
3 

3 

»   -3( 

a 

3. 
2a 

Q   a 
o.  —  > 
c 

3a. 

7c* 

9.1, 
a 

_6. 
a 

10_  4 

}^  42.  7  -^  ^  ■ 
11  6 

44.  -  1,  -  3.       45.  lifVe.       46.  il^^^T. 

Exercise  93. 

10.  ^,    -  ^.  18.  a  +  1,  a  -  1. 
2a          a 


19.^^2,    b  +  2, 

2  2 


11.  a,  1. 

12.  A,    _36.  ^  5 
2a          2a  ^0.  ^^^,    -  ^-^. 

13.  3a,  -  2. 

1^        &        5  21.  -— -»    -— -• 

14. >    — -  •  a6^       a^6 

a      3a 

15        6,        a.  22.  «±-^    -^. 
15.  -->    -^  c         a  +  6 

16.a,-l.  23.-6,«^fA_-. 


a 


a  +  6 


17.  a,  --^-  24.  1,  -^• 

a+1  ate 


ANSWEBS.  XXXI 

Exercise  94. 

1.-1,-7.  8  A,    _i.  14.-1,  f. 

2.  12,  -  7.  '  Sa         a  15.  2,  -  ^. 

3.  f,  -  f.  9.  =1=2,  =t2l/^^.  16.  =1=2,  7,  f 

4.  V-,  -  !•  10.  2,  -  1  =1=  1/"=^.  j7   J    -  3  =1=  1/-  15 

5.  f,  -i  11.  =tl,  ±2.                       *    '              6            * 

6.  3,  f  12.  1,  ±  1,  ±  V^^.  18.  1,  3,  -  4.  _ 

7.  I,  -|.  13.  i  -i  |.  19.  -1,  ±^l/2. 

20.  W,- 3.  21.^1,L±^,Z.1A1^. 

Exercise  95. 

1.  ±1,  i4.      2.   ±1,  if.      3.  1,  I,  .=J-±i^:Z3,  and  "  ^  ^3^"^' 

4.  16,  ^V  7.  1,  ^V  10.  1,  W.  13.  1,  (-  1)1 

5.  1,  f  8.  27,  -  i.  11.  1,  i^%.  14.  i  jI^. 

6.  8,  -  ^.  9.  t,  T^5.  12.  8,  -  xb.         15.  4,  -  6. 

16.   =t:3,  ±^1/2.  23   1    -8    -  7  =^  3l/^  . 


2 
18.  A  ^H-.  „.    .    ,    8  --t  21/37 


17.  ±iV^3,  =tfl/"=^. 


25   5   i 

19.  2,  6,  -  2  i  21/"=T.  ■    '  "  3 

20.  6,  -  ^^V.  ^^-  ^'  ^'^• 

_  27.  i|v/6,  ±81/^:T. 

21.  2,  -  I,  3^J^^ .  28.  -  ^V,  f  1^1^. 

^  29.  I,  -f,  l=tV^ 

00   1    _  5    -  3  ^  31/^.  30.  ±1,  =t^V^310. 

'       ^'  4  31.  =1=21/2,  ^\V^^. 

Exercise  96. 

1.  3,  2.  3.  3,*  12.  5.  8,  ^^*  7.  3,  -  -»/.* 

2.  1,  -  \.  4.  6,  f*  6.  9,  -  f  *  8.  4,  -  f 

9.  2a^  -  2M.  11.  2,  -1,  ^^^'   -l±l/^. 

lO.a^  +  1,  81«!^1.  12.  3,  ¥. 

'9  13.  5,  -  tV 

14.  0,  5.  15.  8,  ^V     ,        16.  i,  -  H-  17.  4,  -  yV 

18.  ±1,  =t|.  19.  ±a,  ±2a2.  20.  2,  f . 

*  Will  not  satisfy  the  equation  as  it  stands. 


xxii 

ANSWERS. 

Exercise  97. 

1.1, 

-6. 

3.  i  -  f .             5. 

f ,     - 

-  I             7.  2,  -  |. 

2.  1, 

-|. 

4.  f ,  -  2.              6. 

1,     - 

-1.             8.  ii,  ^|. 

.M 

a 

'    -2* 

13.  \h  if. 

14.  i,  -  -¥• 

18.      ^     »       1. 
1  -  a 

-^ 

a 

15.  tI^,  -  ¥• 

16.  Y,  -  1. 

19.  ."^    '        \- 
6  -  a     a  +  b 

11.  ±3, 

12.  i2. 

i2l/-2 

•           '''  «'    2I  • 

20.   «  ,      ^^  . 
26       4a 

Exercise  98. 

• 

1.  1, 

2.  -] 

-I 

3.  -  2,  |.             5. 

4.  3,  -  \.             6. 

9  3a, 
2 

-2a. 

11.  a,  ^  • 
a 

14.  1,  f. 

15.  i|,  ±i. 

10.36, 
a 

46  . 
3a' 

12.  27,  -  -2/. 

13.  -^^,  V-. 

16.  ±2,  iii/s: 

17.  6,  -V-. 

18. 

c{c  +  d) 

d(c  +  d) 

21. 

3.7320,  0.2680. 

c-d   ' 

d-c 

22. 

1.41202,  -  0.07868. 

19. 
20. 

3,  f. 

/TT  4-  5i          /^ 

-  3 

r    4-    R 

23. 
24. 

4.67945,  0.82055. 
0.46332,  -  0.86332. 

Exercise  99. 


1.  I,  -  i 

2.  -»/,  -  f . 

3.  1,  16. 


11.  ^v,  -  -V-. 


4.  0,  i4. 

5.  9,  ^f. 

6.  3,  -  2. 


12.  0,  3, 


3i  3l/^=^ 


13.  1,  T?,V 

14.  hS  -  3. 

15.  ^l^i^ZI. 

2 

16.  -1,  -3,   -2±  iV^^^IO. 

17.  15,  -  If 

18.  4,  4. 


0,  A. 

,  1. 

9.^    - 
a 

a, 
6 

^'a 

10.  3,  i. 

19.  - 

1,  - 

3, 

5  ±  1/-  23 
4 

20.  ^ 

+  6 
a 

' 

a-6 
6 

21.  16 

1,  1. 

22.  - 

a,  - 

6. 

23.  2, 

648. 

24.  3, 

-!, 

- 

3  il/43 
4 

- 

25.  - 

2,f 

Exercise  100. 

J1.JV.A.11 

1.  5,  6.            2.  7,  8. 

3.  2,  5.            4. 

5.            5.  23  years. 

6.  30  mi.  an  hr. 

14.  7,  11. 

22.  24. 

7.  2i  3i. 

15.  14,  3. 

23.  4  hrs.,  10  min. 

8.  5,  13. 

16.  10  boys. 

24.  32. 

9.  36  boys. 

17.  9  men. 

25.  5  mi.  an  hr. 

10.  30  and  45  min. 

18.  4i,  7i 

26.  45  mi.  an  hr. 

11.  30  cents. 

19.  18  feet. 

27.  A,  20  days. 

12.  6,  9. 

20.  1  rod. 

B,  12  days. 

13.  6,  15. 

21.  2i  rods. 
Exercise  101. 

28.  9  mi.  an  hr. 

1.  a;  =  1,  -  13. 

6.  X  =  -  2,  -  f  |. 

11.  a;  =  1,  10. 

y  =  2, 16. 

2.  a;  =  4,  -  1. 

y  =  h-  iz' 

7.  a;  =  3,  -  -1^. 

y  =  -  f ,  -  f . 

12.  a:  =  1,  -V. 

^  =  i  -  2. 
3.  a;  =  1,  2. 

2/  =  -  1,  H. 
8.  a;  =  -  3,  7. 

y  =  -  3,  -^/. 
13.  x  =  Z,\. 

y  =  0,  -  3. 
4.  a:  =  2,  -  26. 

y-  -1,  4. 
9.^a:  =  |,  2. 

y  =  -  2,  1. 

14.  a:  =  2,  -  f 

y  =  i,  15. 

6.  a;  =  4,  -  f . 

2/  =  1,  1. 
10.  a;  =  7,  12. 

y  =  \-  f  f 

15.  a;  =  3,  Jf . 

y  =  1,  -  if. 

Exercise  102. 

y  =  2,  3. 

1.  a;  =  ±4,  ^14. 

5.  a:  =  =i=5,  =±=61/2. 

8.  a:  =  ±8,  =t3. 

y  =  il,  ±4. 

2.  a;  =  ±  3,  =t  f  v^. 

.V  =  =f1,  TiV^. 

3.  a:  =  ±],  i|l/-2. 
2/  =  ±2,  ^fl/-2. 

4.  a;  =  ±3,   ifVlTO. 

y  =  ±2,  ±ii/ro. 

y  =  ±2,  =1=71/2. 

6.  a;  =  =i:  1,  ±  Jl_. 

1/91 

y  =  ±3,  i-^. 
1/91 

7.  a;  =  =4=3,  =tf. 

y  =  i5,  =tV-. 

y-=F5,  ±5. 

9.  a;  =±5,  i -i^ 
Vh\ 

2/  =  =f3,  i-^ 
V51 
10.  a;  =  ±l,  i2. 

y  =  ii,  ±|. 

11.  a;  =  ±2,  =4=1/2. 

12. 

a;  =  ±l,  =bl4f. 

y  =  =t  4,  ±  31/2. 

y  =  T3,  ±3f 

Exercise  103. 

1.  a:  =  9,  4. 

3.  a:  =  -  3,  -  7. 

5.  a;  =  =±=7,  ±3. 

y  =  4,  9. 

y  =  -  7,  -  3. 

y  =  ±3,  ±7. 

2.  a;  =  4,  -  3. 

4.  a;  =  4,  -  5. 

6.a:  =  ii  =t|. 

y  =  -  3,  4. 

y  =  -  5,  4. 

y-^h  =ti. 

:xxiv 

^JV^STTEIJ^: 

7.  ar  =  f ,  -  |. 

12.  x  = 

=  =t2a,  ±3a. 

17.  a:  =  f,  2. 

y=-h  I 

y  = 

:  =t  3a,  i  2a. 

y  =  2,  f 

8.  2;  =  2,  1. 

13.  x  = 

=  a  +  1,  a  -  1. 

18.  x  =  i,~l 

y  =  1,  2. 

y  = 

a  -  1,  a  +  1. 

y  =  -h  f 

9.  a;  =  4,  -  3. 

14.  a;  = 

:2,    3. 

19.  a:  =  ±|,  =^f 

y  =  -  3,  4. 

y  = 

3,  2. 

y  =  Tl,  ±f. 

10.  a;  =  7,  -  5. 

15.  a;  = 

=  6,  2. 

20.  a;  =  f ,  |. 

y  =  -  5,  7. 

y  = 

=  2,  6. 

y  =  -  1,  -  i 

11.  a:  =  if,   Tf. 

16.  a;  = 

=  ±i  ±i. 

21.  a;  =  5,  -  3. 

^  =  =ff,   if. 

y  - 

^=ti   ±i. 

2/  =  3,  -  5. 

22.  x  =  a  -  26,  -  2a  -  6. 

23.  a;  =  ±  1  i  1/2: 

y  =  2a  +  6,  26 

-  a. 

2/  =  ±  1  =p  V^. 

Exercise  104. 

1.  a;  =  ±  5. 

4.  a;  = 

2,  -¥• 

7.  a:  ==  1,  |. 

2/  =  i2. 

y  = 

h  -  ¥. 

y  =  3,  2. 

2.  2;  =  1,  i 

b.  x  = 

6,  -5. 

8.  a;  =  ±1,  =b2. 

y  =  f ,  f . 

y- 

-5,  6. 

y  =  i3,  il. 

3.  a:=i4,  ±|V^. 

6.  a;  = 

±4,  ±1. 

9.  a;  =  3,  ^. 

y  =  ±l=F|v/2. 

y  = 

=f2,  =f3. 

2^  =  2,  f  J. 

10.  a:  =  2,  -  f . 
y  =  3,  -  H. 

17.  a;  =  2, 

3 

11.  a:  =  5,  13. 

y  =  l, 

2    -3iT/33 
9 

2/  =  3,  i 

12.^-1,1,7=^^^. 

18.  a;  =  3, 

-  2,  -  2  ±  1/^. 

3/  =  -  2,  3,  -  2  ^  1/5: 

19.  a:  =  ±4,  ±fl/35. 

13.  a:  =  2,  1,  6,  -  f. 

y  =  il,  i^i/S5. 

2/  =  3,  1,  -  10,  -  f. 
14.a:--5,  -1,  7^^-^^. 

20.  a;  =  2, 

,    3±l/-65 
2         * 

y  -  -  2,  -  6,  ^ 

2 
1/-23 

y  =  h 

0    S=FV-6b. 
2 

15.  2:  =  f ,  |. 

2 

21.  a;  =  3, 

0    l±3l/-3 
"  ^'           2        • 

y  =  -  i  -  f 

16.x  =  i  1,-3^^^-33. 

y  ==  - 

2,  3,  l-i3l^E?. 

y-|,j,   -'1^/33. 

22.  a;  =  3, 

-  1,  1  ±  1/-  10. 

18 

y  -  - 

1,  3,  1  =F  V^Tt). 

ANSWERS.  XXXV 

23.  «  «  1,  4, >  y  =  -  4,  -  1, 

24.  a;  =  ±2,  =tl,  ^2V^,  iv/^T. 
y  =  ±l,  i2,  =fV-l,  =f2l/^n[. 

25.  jc  =  3,  -  2,  1,  -  3.  26.  a;  =  3,  4,  -  2  i  V^. 

^  =  2,  -  3,  3,  -  1.  2/  =  4,  3,  -  2  :,=  l/S: 

27.  a;  =  ±  2a,  ±  26.       29.  a;  =  5,  /y.  31.  a;  =  3,  4. 

y  =  i6,  ±a.  y=_i,  1^.  2/ =-1,0. 

28.  X  =  i  f  30.  a;  =  3a,  -  9a.  32.  a;  =  J.  -  7 
y  =  i  f                         2/  =  2a,  -  7a.  y  =  |,  I 

33.  ar  =  -  3,  -  4,  6  ±  V^.  37.  a;  =  1,  V,  2,  5. 


y  =  -  4,  -  3,  6  =F  1/43. 


2        5  1  10 


14  y  =  —  •  '  - 

34.  a;  =  -  >    -  •  a      7a          a         a 

a      a 

L,    _i.  38.  x  =  64,  1. 


6  6 


35.  a;  =  6a,  ^ 
6 


y  =  1,  64. 


36      356  '  39.  a;  =  16,  9,  ^l 

y  =  %  16,  (--»#. 


^=2'       4 


a 
y  =  a  +  6,«(6^. 


2/  =  M,  0. 


41.  X  =  ±4,  ±2,  =tl/^=^/  ±  I/-  V^. 
y  =  i2,  ±4,  il/'^=^/TV~^ 

42.  a:  =  8,  2.  43.  a:  =  4,  3,  6,  2. 

y  =  2,  8.  y  =  f,  2,  1,  3. 

Exercise  106. 

1.  3,  7.  6.  15  by  18  rods.  9.  18c.  @  $12.      > 

2.  3,  8.  6.  28.  or  16c.  @  $14.  i 

3.  7  by  12  ft.  7.  9  by  18  in.  10.  f . 

4  7  by  11  rods.  8.  40  yds.  11.  40  and  60  mi. 

12.  Bow  6  mi.;  stream,  2  mi.  an  hour.  13.  21  and  13  ft. 

14.  10  and  12  ft  15.  $6.  16.  f  and  f. 


XXXVl 


ANSWERS. 


Exercise  106. 

11.  a;»  -  5a;  +  6 

=  0. 

25.  2a;2  -  4a;  +  1  =  0. 

23.  a;»  -  2a;  -  1 

=  0. 

26.  2ar^  -  2a;  +  1  =  0. 

24.  a;'  +  6a;  +  6 

=  0. 

28.  4a;2  +  a^  +  4c^b  =  4aa;. 

Exercise  107. 

1.  Real;  uneq. 

15.  i 

19.  - 

¥• 

23.  -  2. 

2.  Real;  uneq. 

16.  i. 

20.  i 

10. 

24.  3,  -  4. 

3.  Real;  eq. 

17.  |. 

21.  - 

I 

25.  -  3,  -  |. 

4.  Imag. 

18.  V-- 

22.  i 

f 

26.  -  1,  V. 

Exercise  I08. 

If 

13. 

a 

23. 

0,  1,  -  f. 

2.|. 

a-1 

24. 

0,  5,  20. 

3  3a -26. 
•  4a  -  36 

14 

25. 

X 

25. 
26. 

0,5. 
3,  -  1. 

4.  3;  2. 

15 

36 

27. 

3aK 

5.  6a'6. 

6.  2f. 

7.  a'  -  x". 
g  3a.  +  4. 

3a; +  5 

16. 
17. 

oo. 

(a-l)«. 

a-1    . 
a{a  +  1) 

28. 
29. 

a;=-3;y=-4. 

\                 '  a  +  1 
h  =  a  +  l,l. 

18. 

2,  -|. 

42. 

4  and  10. 

9.  11/3. 

19. 

5,  -4. 

43. 

5  and  11. 

10.  66c. 

20. 

-  7,  -  -V-. 

44. 

13  and  19. 

11.  3y. 

21. 

±3,  -2. 

45. 

4  and  8. 

12.  f 

22. 

0,  -4. 

Exercise  109. 

1.  a;  =  1,  5. 

3.  a;  =  15,  8,  1. 

5.  a;  =  a 

y  =  14,  7. 

S/  =  h 

6,  11. 

y  =  7. 

2.  a;  =  3. 

•  4.  a;  =  9. 

6.  a;  =  8. 

y  =  2. 

y  =  4. 

2/  =7. 

7.  a;  =  5,  12,  19,  etc.  11.  99,  75,  51,  27,  3. 
y  =  2,  7,  12,  etc.  8,  32,  56,  80,  104. 

8.  a;  =  7,  37,  67,  etc  12.  243,  126,  9. 
y  =  1,  14,  27,  etc.  78,  195,  312. 

9.  a;  =  3,  14,  25,  etc.  13.  V,  -\S  |,  f . 
y  =  2,  18,  34,  etc.  j%,  |i,  f |,  f f. 

10.  X  =  22,  57,  etc.  14.  Sheep,  3,  14,  25. 
ff  =  10,  23,  etc.  Calves,  16,  10,  4. 

16.  12  ways.*  16.  11  ways.*  17.  159;  an  indefinite  No. 


*  Including  zero  yalues. 


ANSWEBS.  MXVU 

Exercise  llO. 

1.  15.  5.  ±2.  9.  4.  13.  9,  -  f . 

2.  2\.  6.  f  10.  8.  j4    3^,    _5. 

3.  5.  7.  ^V2.  11.  a;  =  -  2y.  "  ^    '       a? 

4.  8.  8.  35.  12.  z^  -2x  +  2^y^  -2y. 

15.  ^.  18.  10291  ft.  21.  157}. 

16.  ^.  19.  13  in.  22.  4. 

17.  «;  =  f  -  4ar»  +  -\  •  20.  12  in.  23.  9(  ±  VG  -  2). 

Exercise  111. 

1.  31.  5.  54,  94^.  9.  -  69. 

2.  -  25.  6.  i,  0.  10.  -  189. 
«=-81.                           8  =  3|.  11.  148|. 

3.  -  46.  7.  -  7.2.  12.  -  300. 
s=  -  364.                          »  =  -  37.8.  13.  573^ 

4.  _  23^  -  13.  8.  164.  14.  -  165. 

15.  (^"^  -56)6c.  17.-771/3. 

4 

Exercise  112. 

1.  a  =  4  ;  s  =  286.  9.  n  =  21 ;   d  =  1. 

2.  a  =  -  5f  ;  s  =  209.  10.  n  =  21 ;  d  =  -  2. 

3.  a  =  5  ;   d  =  4.  11.  n  =  18  ;   d  =  -  ^Jj* 

4.  a  =  11 ;   d  =  -  3.  12.  n  =  16  ;   d  =  f. 
b.  a  =  bl,  d=  -2^.  13.  a  =  7  ;  n  =  6. 

6.  a  =  f  ;   d  =  -  i.  14.  a  =  -  5 ;  n  =  10. 

7.  a  =  -  I ;  rf  =  tV  15.  a  =  1 ;  n  =  7. 

8.  a  =  3f;d=-i  16.  a  = -i;  n  =  16. 

17.  12.  18.  8.  19.  10.  20.  4  or  9. 

Exercise  113. 

1.  d  =»  -  2.  3.  d  =  f  5.  d  =  f .  7.  a:.  _ 

2,  d  =  i.  4.  d  =  -  xV  6.  -  Hi.  8- 


^-y* 


Exercise  114. 

1.  5,  7.  5.  n\  8.  8,  7i 

2.  4i  3.  g   7n(n  + 1) .  9.  3,    -  2,  -  7,  - 11 

3.  -  -V,  -  ¥•  '  2  10.  1,  3,  5,  7, 

4.  5c  -  7&,  4c  -  e&.  7.  X02d  t«rm.  11,  24d, 


xxxviii 

^ivsTrj5;i25'. 

12.  1,  4,  7  . 

13.  12.               14 

.  ~  5,  -  41 

..     fll,  6,  1,  -4,-9. 

21. 

Exercise  115. 

1.  486. 

4.  16.                     6.  32. 

10.  mi 

2.  192. 

8  =  55.             7.  -  63. 

11.  -V(3  +  1/5). 

3.  -  i^. 

5.  j\.                    8.  -\V-. 

12.  31(1/6  +  1/S^) 

s  =  33if. 

Exercise  116. 

13.  211/^  +  28. 

1.  2. 

4.  45. 

7.  -  4.             13.  5. 

8  =  728. 

8  =  -%W. 

8.  I                14.  6. 

2.  5. 

5.  -  ^. 

9.  -  f.             15.  5. 

»  =  -  425. 

8  =  -  H¥^- 

10.  -  I            16.  4. 

8.  16. 

6.  f. 

11.  5.                 17.  6. 

,-ii^. 

65  +  191/6^ 

12.  6. 

Exercise  117. 

1.  r  =  i  3.  r  =  -  2.  5.  r  =  -  ^  7.  =t  f . 

2.  r  =  |.  4.  r=-4.  6.  r  =  1/2.  8.  ±5. 

Exercise  118. 

1.  3.  6.  i  12.  ^j.  17.  3t\^V 

2.  f  7.  H.  13.  H|.  18.  Iji^. 

3.  -  ^.  8.  6(2  +  1/2).  14.  5xVt.  19-  lyV 

4.  ^.  9.  K31/2  +  4).  15.  3Hi  20.  |Mf 
6.  W.                11.  a.                              16.  Un. 

Exercise  119. 

1.  f,  f 13.  3,  6,  12.  jg     r2,  4,  8,  12. 

2.  A,  f 14.  1,  3,  9,  27.  ■   ^-¥-,  ¥,  I  I 

3.  ¥,  V- .5    8    11  19-  -24|,  f. 

4.  96,  48 "^^^   i  15    8    l'  ^0.  $64,  80,  100,  125. 

11.  5,  15,  46.                                  '     '     •  21.  27,  36,  48  yrs. 

12    /7,  14,28.  16.115,30,60,120.  r  2,  4,  6,  9. 

•    1 63,  -  21,  7.           17.  21/2  +  3.  ^  2,  i,  -  f ,  9. 
Exercise  120. 

6.  -  Jy^,  -  H,  etc.  jQ       a;j/     ^       xy     . 

7.  ff  '  2x  -:i/     Sx-2y 

8.  -  36.  11.  12  and  18. 

9   ^Lui.  12.  -i,  -i  -i 

««  +  1  13.  8  and  32. 


1. 

A. 

2. 

-A- 

3. 

tS. 

4. 

1. 1,  h  !• 

5. 

f ,  1,  A,  et<x 

ANSWERS.  xxxix 

Exercise  121. 


1.  360.                 6.  24,024. 

2.  167,960.          7.  2,441,880. 

3.  2520.               8.  43,758. 

4.  462.                 9.  15,120. 

5.  969.               10.  1,961,256. 

11.  7,893,600.              16.  2520. 

12.  286.                        17.  21. 

13.  40,320.                   18.  56. 

14.  4536.                      19.  8,877,690. 

15.  1st,  60,480;  2d,  181,440. 

Exercise  122. 

1.  60.                         7.  168,168. 

2.  3780.                      8.  23,000. 

3.  453,600.                 9.  7560. 

4.  34,650.                 10.  3,531,528 

5.  1,663,200.            11.  15,840. 

6.  1,135,134,000.      12.  302,400. 

13.  1,596,000.        19.  4845. 

14.  420.                  20.  210. 

15.  36.                    21.  5040. 

16.  2880.                22.  31. 

17.  8640.                23.  72,000. 

18.  1200.                24.  27,720. 

Exercise  123. 

1.  2.                            3.  -  4. 

2.  -  15.                       4.  -  8. 

5.  6  =  c{c  +  a). 

6.  c  =  d{h  -  ad). 

7.  ^  =  d  -  a  =  1.               8. 
d 

^      ^  _d  —  hn  +  n^  _  dm  -  en , 
mn                  v? 

Exercise  124. 

l.l  +  x-x'  +  x^-x' 

2.  2  +  a;  +  2a;2  +  4ar»  +  8a;* 

S.  1-6X  +  Ix"  - 
4.  3  -  Ix""  +  14a;* 

13a;3  +  20a;* 

-  28a;«  +  56a;8 

5.  l-2a;  +  2a:»-2a:*  +  2a;«... 
,6.  a;  +  5a:2  +  7a:3-a:*-23a;5..., 

. .    7.  1  -  a;  -  a;'  -  la;*  +  }a;6 

..     8.  l  +  ia;  +  ia;^-fa;»-Ha;»..... 

9.  ^  -  2a;  +  ^-a^'  - 

10.  f  -  -V-a;  +  i\x' 

11.  f  -  fa;2  +  -V-a;' 

_  n^.^  +  4 993.4 

--W-^  +  Mf^* 

+  fa;*--2a;* 

12.  x-»-3a;-i  +  3-3a;  +  3a;2... 

13.  2a;-H2-a;-3a;'-2x3 

14.  2x--^-^\-\x  +  ^x'-^:x?... 

. .     15.  a;-3-2a;-Hl-|a;  +  ia;» 

16.  a;-'»-a;-'-2a;  +  2a;'-4a;» 

..     17.  a;-''  — a;-^  +  a;-a;'+a;* 

Exercise  125. 

l.l-x-\x^-'\:x?-^x*.... 

2.  1  +  2a;  -  a:*  +  2a;3  -  f  a:* . . . . 

3.  2  -  fa;  + /^a;^  +  A^.a;' 

•  4.a       ^'        ^          ^    

2a      8a'        160*^ 

•  5.  1  +  ia;  -  \x'  +  /i-a;» 

6.  1  -  a;  -  3a;»  -  ^7? 

ANSWERS. 
Exercise  126. 

1   1-.  ^     .  8.x-'-x+l+ ^ ^— 

X     x-Z  3(a;  -  1)      3(x  +  2) 

x-2      x  +  l  2x      3a; -2      3a; -1 


x-S      a;  +  3  x-l     a;+l      2a;- 3      2a;+3 

^   1^  _       4      ^      3      .  jj       3a 2a      . 

'  X      2a; +  1      2a;-l  'a;  fa       a;- 4a 

6.  a;  +  3- -^ +— ^  •  12.  a;  +  2a  +       "'              "' 


2x      X—  2                                        2x  -  a      X  +  a 
6.  i  +  — —  •  13.  a;  -  2  + +         ^ 


X      2a; +  1      a;-l  a;  -  2K2       a;  +  21^2 

7.  a;-6+-A----A-^.       14.  ^ +  ? ^ 

4a; -3       3a; +  2  x-2  +  V^2      a;- 2 -1/2     ^ 

1  +  V2  .        1  -  1/2 


15. 


2a;  -  5  +  1/2       2a;  -  5  -  1/2 
Exercise  127. 


1.  l_i_  +  _A_.  4.3  +  5  9 


a;       a;'       a;  +  1  a;       a;'       3a;  —  4 

2.-^ g—  +        1        ■      5.  1  5 


a:+l       (a; +  1)2       (a;  +  1)'  3(1 -2a;)     3(a;-2)     (a;-2)2 

3      ^       12 10_ .     g        7       +  _     12  .         5 


x  +  2      (a; +  2)2       (a;+2)^  2a;  -  1      (2a;  - 1)^       (2a;-l)» 

7    2^  _  ^ 1_  10 


X       x^       7?      5a;  —  1 
8   _5 2        ^3  13 


a;  -  1      (a;  -  1)2      a;  +  2      (a;  +  2)" 
9.-J-+        6        +       12       ^        3 


a; -2      (a; -2)2      (a;  -  2)=«      (a; -2) 

10.  1  -  A 10_  ^        48     _  . 

X       a;2       (2a;  +  3)2       (2a;  +  3)' 
ll.:.-2-,-— 6 X_^        4 


2a;  +  3      (2a;  +  3)2       a;  -  1      (a;  -  1> 
12.1-A__JL_+         4 


X       7^      a;  +  1      (2a;  -  5)^ 


ANSWEBS,  xli 

Bxercise  128. 

.    3a;  -i-  5  _  3^  _  J5_. 
'   x^  +  I        X      '  X* 

%-^ ----■*      .  5       ^-^  a;-6     . 


1 

a; 

2a; -5. 

a;='  +  2 

3              3a;- 

4 

a: 

-2      a;*  +  2a;  +  4 

10a;-  7 

4 

Z.—±}^±^-L 3 6. 


a;'  +  a;  +  l      a;»-a;  +  l 
a;  +  5       .       2a;  -  15 


3(a;»  +  a;  +  1)      3(a;  -  1)  a;'  -  a;  +  3      (a;*  -  a;  +  3)' 

2a;  +  3  2a;  -  2  1 


3(2a;'^  -  3)       (2a;''  -  3)»       3a; 
2a; -1         a;  +  2  3  1 


a;2  +  1       (a;»  +  1)'       2a;      2(a;  -  1) 

Exercise  129. 

1.  «  =•  y  -  2/'  +  y'  -  2/* 5.  a;  =  y  -  ^y»  +  j2/»  -  |y*. . 

2.  a;  =  y  +  3^2  +  13y»  +  67^/* 6.  a;  =  2^  +  l^/'  +  ^y'  +  ffty*- 

3.  a;  =  y  -  2?/2  +  5^/'  -  14^ 7.  a;  =  \y  -  W  ^  i^y^  -  ^^jV^ - 

4.  a;  =  y  +  i2/'  +  i3/=«+^V2/* »•  ^  =  y-iy'  +  ihy'-h^y' -- 


^3         15         315 


10.  a;  =  -  My  -  1)  +  1(2/  -  ir  -  ^(2/  -  1)' 

11.  a;  =  (2/  -  1)  -  K2/  -  1)'  +  1(2/  -  1)'  -  i(2/  -  1)* 

Exercise  130. 

1.  a*  +  12a'  +  54a=  +  108a  +  81. 

2.  32a5  -  80a*a;  +  SOa'a;'  -  40aV  +  lOaa;*  -  x^, 

4.  81a;2  _  216a: V  +  216a;i/*  -  96a;  V  +  162/®- 

5.  a;^  -  lOx^  +  40a;'  -  80a;"  +  80a;'^^'  -  32a;*. 

6.  a^y^  +  Sx''y^  +  2Sx^y^  -f  56a;5^^  +  70a; V  +  56a;V^  +  28a;'2/+8a;2/^  +  l. 

7.  a;""^^*  +  7a; ~  ^  +  21a; "  "^  +  35a;~ ^  +  35  +  21a;^  +  7a;^  +  x^ 

8.  ^^MZL  _  ^5^a;ly -i  +  ^x*y - '  -  ^x^y ~^  +  |a;V  -  ^^y^- 


xlii  ANSWEBS. 

10.  243a^a;~^  -  405a'a;-«  +  270a^x~^  -  90aa;-'  +  15aK~^  -  1, 

11.  16a:«  +  32a;"'^V  +  24a:^y^  +  8a;  V  +  ^^M- 

12.  a*  -  6aS^  +  15a~^  -  20a~^  +  15a~'^^  -  6a~^^'  +  a-». 

13.  a^x^  -  12a^x^  +  54a  "V  -  108a  ~V^^'"  +  81a- V. 

14.  64a*a; - ^ 4- biea^^x ~  ^  +  2160a ~^x~^  +  4320a "  ^x^  +  4860a ~  ^^x^ 

+  2916a  ""^'a;'^'  +  729a -»a:». 

15.  Sla-W-  108a-26-2  +  54a-i6-«-  126-^"  +  a6-i*. 

16.  a^-Zx^  +  9x*-  IZx^  +  ISx^  -  12a;  +  8. 

18.  16x»  +  32a:^  -  72a:«  -  136a;*  +  145a;*  +  204a;'  -  162a;2  -  108a;  +  81. 
20.  81x8  _  216a;'  4-  108a;«  +  120a;5  _  74^^*  _  403^3  ^  I2a;2  +  8a;  +  1. 

21.  -  14,784aV».  24.  -  198a"*3V.  26.  -  61,236aTV. 

3  2  27.  —  1320a;^. 

22.  1716a;Vi  7920a^-6*.  28^  1365a*a;i8. 

23.  3003a;"y«.  25.  i-%^^-5.x-^yK  ^9.  ^f f^x^*. 

J  30.  -  212,640. 

1001a;  ^yio.  mi^x^yn.  31.  24,310a;2/". 

Exercise  131. 

1.  a-»-  2a-3a;  +  3a-*a;2  -  4a-5a;3  +  5a-«a;* 

2.  ar»  -  fa;  +  fa;-*  +  ^^x-^  +  jh^'^ • 

3.  a;^+  3a;-3_|a.-f  ^  f |a; ~ ^  -  f §f a; ~ ^ 

4.  1  -  2a;'  +  4a;*  -  8a;<  +  16a;8 

5.  a;-«  +  3a;~  *^'  +  6a;-9  +  lOa:""''^^"  +  15a;-" 

6.  a;^  +  |a;~^- fa;~^  +  ffa;"^-^H^~^ 

7.  x^  -  \xS^  +  \x^y  -  ^^x-^y^  +  A\a;'^V 

8.  1  -  7a;  +  28a;2  -  84a;3  +  210a;* 

9.  a^  -  \a-^x  -  j\a-^x^  -  jha"'^'^  -  ^iha''^x* 

10.  a-a-^x-  a-H^  -  !« -s^  _  1^0^-113.4 

11.  a;^  -  14a:  V  +  "^Oa;^  -  140a;  V  +  70a;  ~  ^2/" 

12.  a ^  -  4a^» ~ .^  +  lOa'a;-'  -  20a'a; ~  ^  +  35a'^^a;-« 


ANSWERS.  xliii 


13.  a: "  ^  +  ^ax  ~  ^  +  f  a'a;  ~  ^  +  ||a»a; ""  ^^'  +  if  f  a*a;  ~  "^ 

14.  a;  - 1  +  f  a;  ~  ^  +  -V-^^  -  *  +  W"^  ~  ^^'  +  -W/a;  -  ^ 

15.  a^  -  a^x^  +  2ay  -  Ya^^  +  -%^-a"^^'x^ 

16.  x~^  -  x^y^  -  ^x^y  -  ^x'^^y^  -  f aj'^V' 

17.  1  +  ix^y  ~  ^  +  '\*x^y  -  ^  +  Vt^^  ~  ^  +  If  f  ^^^  - ' . . . 


18.  x^y    ■*"  -  5a;%    ^  +  V-a;    ^y"^  -  fa;    ^^  *  -  |a;    ^"^/^ . 


19.  ^^jx^. 

24.  Ya^ ;  -  55^V^a;i9. 

20.  -  ^l2^a;^ 

21.  iof<ia  ~  ^a;i 

25.  ^a;-V;  ^Uz^-'V'- 

26.  jih^-''. 

22.  ila-^x\ 

27.-  mx-*. 

23.  |a:^;iMa:"- 

28.  a'^x^l 

29.  3.16228. 

31.  3.10723 

33.  9.695359.            35.  4.01553. 

30.  8.062258. 

32.  5.03968 

34.  1.96799.             36.  2.03616& 

Exercise  132. 

1.2+1          1 
2+      1  + 

1 
1  + 

1  . 
2 

3.5+1        1         11. 
1+2+2+3 

2.3+1          1 

2+      1  + 

1. 
3 

^1111111 
'1+     1+     1+     1+     1+     1+5' 

5.  If 

6. 

¥• 

7.  Y.                8.H. 

9.  2  +  -^ 

2  + 

1 
3  + 

1 
1  + 

1  . 
2 

5th  conv.  =  f  i. 

10.  1  +  -i- 

3  + 

1 
1  + 

1 

2  + 

1 
3  + 

1  •      5th  conv.  =  f  f 

11.  1  +  -^ 

2  + 

1 
3  + 

1 
1  + 

1 
3  + 

-1-    I'      5th  conv.  =  If. 

1  +     o 

12.  .\       1 

1 

1 

1 

1         11-      5th  C- if. 

2+2+      1+      1+      2+      5+      1+     2 

l^'aT'^    TT    I*      0°ly4convergent8. 

14.  4  +  -^    -^    -^     -^    -^    ^  •      5th  conv.  =^  W. 
2+3+4+2+1+2  ^^ 


xliv  ANSWERS. 

IK        1  1  1  1  1  1  1  1  r,U „  11 

15.  — —    -r —    - —    - —    — —    — —    — —    —  •  otn  conv.  =  i*. 

4+S+1+1+1+2+3+4  *^ 

16.  1  +  -^    -^    -^    -^    -^    -^    ^  •  5th  conv.  =  n. 

1+1+2+3+1+1+2  ^^ 

^^'  3T    3T    3T    6T  TT  ¥+   TT   10*  ^^^^«^^-=^V 


Exercise  133. 
1.  2+-4---^  4.  2+1        1 


4+4+ 1+4  + 


2.  X  +  -i--L.  5.  2  +  J--l^-l--l- 

1+2+ 1+1+1+4  + 

3.3+-^-!-  6.  3  +  -i ^ 1 ?- 

6+     6+ 1+2+1+6  + 

7.4+-i-    ^  J-    -^    -JL 

2+      1+  3+      1+      2+  

8.4+-L    J_  _1_ 

2+8+2+  

9.4+1         1  1 


4+      8+      4+ 

1         1         1 

1+      1+      1+      1+      2+ 


10.1+1         1         1         1         1 


'1+3+      5+      2+      28+     2  + 
12.     1         1         1  1  1 


5+      1+      2+      1+      10  + 

13.  7+-i-  J-  JL.    JL 

1+  3+  1+      8+-. 

14.  2+ -J-  -I-  -1- 

3+  2+  3+  

15.  2+  -I-  -I-  J_    _1- 

1+  3+  2+      1+-. 


16.  1^-^lA.                 18.  ^-1  .  20   ^^5^ 

2                                      2  *        2 

17.  V^  ^  2.                     19.  ^^-1  .  21   ^37  +  11 

2  *          4 


* 

ANSWERS 

1. 

X 

22.4./^ 

1 

A  -4-      . 

23.  5+ -L^ 

1 
6+  « 

o  + 

1111 

1+      1+      4+      9+     ■ 

1 
4+ 

r2+  1 

25.   ^            V 

1 

1         1         1 

1         1 

1 

1  + 
.    1 
1  + 

6+      1  +      1  + 

1         1         1 

1+      1+     6+ 

1 +      1+      6+ 

o     o     O     0     C 

26. 3,  -V-,  m, 

355     .    . 

TT3 

. 

27.  27,  -V-,  -W-,  -\W- 



^ 

28.   i,    3,'^,    ^3, 

Nj,  ^ 

655          694,    .    . 

D    ^7"0¥>    ^^55    •   * 

Exercise  134. 

4.  0.7782. 

15.  7.3365  - 

10. 

18.  0.4774. 

5.  1.9542. 

16.  3.1055. 

19.  9.8914  -  10. 

11.  8.9031  - 

10. 

17.  1.8008. 

20.  8.6309  -  10. 

Exercise  135. 

1.  43. 

4.  3.78.                   9. 

.00803. 

18. 

0.4367, 

2.  770. 

8.  0.627.                17. 

283.6, 

19. 

.05925. 

Exercise  136. 

1.  105. 

17. 

1.324.              33. 

3.936. 

49. 

-  2.158. 

2.  34.3. 

18. 

43.36.              34. 

1.946. 

50. 

.02864. 

3.  .0755. 

19. 

-  4.08.            35. 

0.459. 

51. 

-  2.483. 

4.  207.71. 

20. 

0.2785.            36. 

18.02. 

52. 

0.873. 

5.  4.082. 

21. 

0.4287.            37. 

14.44. 

53. 

1.792. 

6.  0.5036. 

22. 

12.16.              38. 

5.624. 

54. 

.01567. 

7.  64.7. 

23. 

37.82.              39. 

-  9.365. 

55. 

-  3.908. 

8.  -  0.7995. 

24. 

162.5.              40. 

0.3933. 

56. 

7.672. 

9.  0.04775. 

25. 

0.7518.            41. 

1.403. 

57. 

0.8686. 

10.  147.1. 

26. 

0.2526.            42. 

2.052. 

58. 

-  0.4704. 

11.  -  2.773. 

27. 

4.359.              43. 

0.1755. 

59. 

-  0.1606. 

12.  -  0.2681. 

28. 

1.487.              44. 

22.58. 

60. 

0.2415. 

13.  0.2168. 

29. 

1.502.              45. 

1.19. 

61. 

.0725. 

14.  1.427. 

30. 

11.86.              46. 

-  1.162. 

62. 

3.076. 

15.  2.407. 

31. 

0.6633.            47. 

3.271. 

63. 

1.805. 

16.  0.3016. 

32. 

2.571.              48. 

0.1424. 

64. 

0.7876. 

Ivi 

ANSWERS. 

Exercise  137. 

1.  1.17. 

5.  7.52. 

9.  1.206. 

13.  3.37. 

2.  1.54. 

6.  0.76. 

10.  2.12. 

14.  -  1.38. 

3.  1.54. 

7.  3.47. 

11.  0.537. 

15.  -  0.24. 

4.  1.65. 

8.  1.88. 

12.  0.81. 

16.  21oga  +  31og6, 

17. 

log  a  +  5  log  X. 

18.  3  log 

6  +  ^  log  X. 

20.  idog 

a  +  log  a;-  3  log  6}. 

22.  ilog 

a  +  1  log x-\o^h  -\ 

log  c. 

25.  ^og7         .  28.  ^og3-log7  4-  21og6  . 
log  a  —  log  6                                               log  a 

26.  log^        .  29.  ^og  CT  +  log  6  -  log  5  . 
log  a  —  log  c  *        log  a  -  2  log  6 

27.  log6-logl3.  30.               logll-log6 

log  13  -  log  a  2  log  6  +  log  c  -  log  (a  —  6) 


log  (g  -f  6)  +  log  (ct  —  6) . 

31og{2a-l) 
I  \  log  fee  +  6)  +  I  log  (ct  -  6)  -  log  25  1  '. 


I  log  a 

33.  3.56.  38.  2.03.  43.  16.  48.  4. 

34.  1.19.  39.  -.065.  44.  25.  49.  f. 

35.  0.71.  40.  -  3.46.  45.  32.  50.  -  f. 

36.  0.84.  41.  1.58.  46.  %.  51.  -  f. 
87.  0.83.  42.  27.  47.  3.  52.  -  4. 

61-64.  See  page  326,  formulas  17-20. 

65.  7.  67.  9.  69.  $2497. 

66.  8.  68.  $2654.  70.  14.2  and  10.24  yrs. 

Exercise   142. 

25    0.  28.  -271.  35    _^!jt^'  37.  —6f. 

26.44i.  29.25.  '  ^"^"     "         38.-%- 

2a;2  ft  2 

27.  i.  34.  -x-^iw/x-  14.    ^^-  2^r^2-  39^  |^e; 

^Q    27  i/6  +  99  1/2"—  48  i/3  — 176 
*  94 


u 


i 


11  iiiiiiiiii  iiiiii  iii^ 


